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QP461  .M57  An  introduction  to  t 


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Volume  II 


SCIENCE  SERIES 


Number  i 


THE 


UNIVERSITY  OF  MISSOURI 
STUDIES 


EDITED  BY 
\V.    G.    BROWN 

1'rofess.or  of  Chemistry 


AN  INTRODUCTION  TO  THE 
MECHANICS  OF  THE  INNER  EAil 


MAX  MEYER,  Ph.  D. 

Professor  of  Expcrimrjital  Psychology 


PUBLISHED    BY   THE 

UNIVERSITY  OF  MISSOURI 
December,   1907 

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I 

AN   INTRODUCTION   TO   THE 
MECHANICS   OF   THE   INNER  EAR 


Volume  II  SCIENCE   SERIES  Number  i 

THE 

UNIVERSITY  OF  MISSOURI 
STUDIES 


EDITED  BY 

W.  G.  BROWN 
Professor  of  Chemistry 


AN  INTRODUCTION  TO  THE 
MECHANICS  OF  THE  INNER  EAR 


MAX  MEYER,  Ph.  D. 

Professor  of  Experimental  Psychology 


PUBLISHED    BY    THE 

UNIVERSITY  OF  MISSOURI 
December,  1907 


Copyright,  1907,  by 
THE  UNIVERSITY  OF  MISSOURI 


COLUMBIA,   MO.: 
E.    W.     STEPHENS    PUBLISHING    COMPANY, 
1907 


PREFACE 

About  two  thirds  of  this  study  has  been  pubHshed  at  different 
times  in  various  German  scientific  periodicals,  chiefly  in  the  Zeit- 
schrift  fiir  Psychologie  imd  Physiologic  der  Sinnesorgane.  The 
author  has  long  hesitated  to  present  in  book  form  the  results 
of  his  labor  in  this  remote  corner  of  scientific  investigation 
because  the  interest  in  these  problems  seems  to  be  neither 
intense  nor  general.  This  lack  of  interest  on  the  part 
of  the  scientific  public,  however,  is  not  due  to  the  unimpor- 
tance of  the  subject,  but  rather  to  a  wide-spread  conviction 
that  all  the  problems  pertaining  to  it  were  solved  half  a  cen- 
tury ago  and  that  therefore  nothing  problematic  is  left.  For 
years  during  which  —  since  his  student  days  —  these  pro- 
blems have  been  in  the  mind  of  the  writer,  he  has  belonged 
to  an  exceedingly  small  minority  of  scientific  men,  who  have 
not  permitted  themselves  to  become  captives  of  this  convic- 
tion. But  since  this  minority  is  gradually  increasing  in  num- 
ber, and  since  professional  friends  have  encouraged  the  writer 
he  has  decided  to  lay  before  the  public  the  results  of  his  in- 
vestigations in  a  continuous  exposition  of  his  theory  as  far  as 
it  goes  at  present.  It  is  natural  that  he  has  preferred  to  do 
this  in  the  English  language,  since  nearly  all  his  previous 
publications  concerning  it  are  in  German. 

The  author  does  not  pretend  to  present  in  this  book  a 
complete,  perfect,  and  final  solution  of  the  problem  concern- 
ing the  mechanics  of  the  inner  ear.  His  farthest  reaching 
hopes  will  be  fulfilled  if  he  succeeds  in  impressing  upon  the 
reader's  mind  the  fact  that  there  are  here  still  problems  left 
for  solution  and  in  giving  these  problems  such  a  clear  and 
definite  formulation  that  the  interest  of  others  will  be  turned 
towards  them.  There  is  little  hope  for  a  final  solution  of 
these  problems  except  by  the  co-operation  of  many  investi- 


gators.  The  contents  of  this  book  are  arranged  from  a  peda- 
gogical rather  than  from  a  logical  poinc  of  view.  The  author 
does  not  intend  to  present  a  systematic  representation  of  his 
own  ideas  for  comparison  with  the  ideas  of  others,  but  rather 
a  series  of  lectures  as  he  would  deliver  them  before  a  class  of 
college  students,  not  presupposing  any  knowledge  or  any  in- 
terest but  what  a  somewhat  advanced  college  student  might 
be  expected  to  possess.  A  reader  who  should  prefer  to  make 
himself  acquainted  with  the  contents  of  this  book  from  an- 
other point  of  viewj,  will  be  able  to  do  this  by  the  aid  of  the 
index  added. 

The  author  has  attempted  to  omit  as  much  as  possible 
everything  of  a  polemic  nature.  His  criticism  of  the  views 
of  other  investigators  may  be  found  in  his  previous  publica- 
tions. In  this  book  he  does  not  propose  to  record  the  views 
of  other  scientists,  but  the  conclusions  which  he  has  reached 
himself  after  more  than  a  decade  of  thought  concerning  these 
problems.  For  the  reader  who  might  be  interested  in  the 
development  of  the  author's  thought  concerning  these  prob- 
lems, he  has  added  at  the  end  of  the  book  a  list  of  those  pub- 
lications of  his  own  which  are  directly  concerned  with  the 
problems  here  presented. 

The  author  hopes  that  this  booklet  will  help  to  break 
down  the  barrier  of  dogmatism  which  has  too  long  stood  in 
the  way  of  progress  in  this  field  of  scientific  inquiry,  and 
which  is  still  far  from  being  a  thing  of  the  past.  It  is  truly 
dogmatism  to  profess  that  the  application  of  so  simple  a 
theorem  as  that  of  Fourier  can  do  justice  to  an  attempt  at 
comprehending  the  mechanical  processes  underlying  the  won- 
derfully complicated  and  unfortunately  only  superficially 
known  phenomena  of  audition. 


THE  MECHANICS  OF  THE  INNER  EAR 

Everyone  knows  that  the  part  of  our  body  which  in  ordi- 
nary life  we  call  the  ear  and  which  anatomists  call  the  pinna, 
is  not  the  organ  of  hearing  but  a  mere  ap- 
The  external  ear  pendage  to  the  organ.  Its  chief  utility 
consists  in  the  fact  that  it  aids  us  in  dis- 
tinguishing sounds  coming  from  a  source  in  front  of  us  from 
sounds  in  our  rear.  We  know  how  much  more  difficult  it  is 
to  understand  the  words  of  a  speaker  behind  us  than  the 
words  of  one  who  stands  before  us.  We  can  reverse  this  con- 
dition by  forming  of  our  hands  leaves  similar  to  the  external 
ears,  but  naturally  larger  and  placing  them  opposite  the  ears, 
that  is  in  front  of  the  opening,  the  auditory  passage.  Then, 
sounds  from  the  rear  can  enter  the  passage  and  reach  the 
tympanum  with  a  much  greater  force  than  sounds  coming 
from  the  front.  Animals,  being  able  to  move  their  external 
ears,  can  use  them,  of  course,  to  greater  advantage  than  hu- 
man beings. 

The     organ   of     hearing — in      the     narrower     sense   of 

the  word — that  is,  the  anatomical  structure  within  which  the 

ends  of  the  auditory  nerve  fibres  receive 

The  tube  con-  their     peripheral     excitations,     is     to     be 

taining  the  sense      ^^^^^  stretched     out     along    the    central 

J       _       ^  line    of    a  tube  which  is  very  narrow    rel- 

and  narrow  ■' 

ative  to  its  length.  This  tube  is  called 
by  the  anatomists  the  cochlea,  because 
it  is  not  built  in  the  form  of  a  straight  line,  but  coiled 
up  like  the  tube  of  a  snail  shell.  The  advantage  of  its  being 
coiled  up  in  this  way  is  obviously  not  to  be  sought  in  its 
mechanic — or  rather  hydrodynamic — function.     At   least,   no 

(O 


2  UNIVERSITY  OF  MISSOURI  STUDIES 

one,  to  the  writer's  knowledge,  has  ever  expressed  himself 
as  inclined  to  look  for  it  there.  For  its  hydrodynamic  func- 
tion it  is  clearly  of  no  great  importance  whether  the  tube  is 
curved  or  straight,  and  we  shall  speak  of  it  in  the  following 
for  the  most  part  as  if  it  were  straight,  in  order  to  simplify 
the  discussion.  The  real  advantage  of  this  shape  of  the  tube 
is  doubtless  a  mere  anatomical  one,  it  being  possible  thus  to 
find  a  better  place  for  it  in  the  base  of  the  skull. 

We  must,  in  order  to  understand  the  function  of  this 
tube,  be  aware  of  the  fact  that  it  is  filled  with  a  watery  fluid, 

lymph,  and  that  its  walls  consist  of  hard 
The  contents  of  unyielding  bone.  Now,  when  we  go 
the  tube,  a  fluid,  through  the  literature  of  the  subject,  we 
is  incompressible      often  see  writers  speak  of  waves  in  the  fluid 

which  are  said  to  pass  along  the  tube  as 
air  waves  move  in  a  tube  filled  with  air.  Views  of 
this  kind  cannot,  of  course,  contribute  towards  an  un- 
derstanding of  the  process  of  stimulation  of  the  periph- 
eral nerve  ends.  They  are  not  rational  considerations  of  the 
facts  before  us,  but  theoretical  dreams,  forgetting  the  physical 
conditions  of  the  case.  Let  us  regard  the  velocity  of  the 
sound  in  such  a  fluid  as  that  of  the  inner  ear  as  about  fourteen 
hundred  meters,  let  us  remember  that  the  whole  length  of  the 
tube  is  only  a  couple  of  centimeters,  let  us  understand,  then, 
that  even  with  rather  high  tones  of  short  wave  lengths — 
beyond  the  musical  range — the  total  length  of  the  tube  is  only 
a  small  part  of  the  spatial  length  of  the  waves  said  to  travel 
up  and  down  the  tube;  and  we  shall,  admit  at  once  that  to 
speak  of  tone  waves  travelling  in  the  lymph  up  and  down 
the  tube  is  like  speaking  of  a  horse  race  which  is  to  take  place 
within  a  dog  kennel.  We  have  to  follow  the  custom  of  the 
physicists  who  in  such  cases  neglect  the  compre^isibility  and 
elasticity  of  the  small  volume  of  fluid  altogether.  We  must, 
therefore,  regard  the  fluid  in  the  cochlea  as  being  of  identical 


MECHANICS  OF  THE  INNER  EAR 


density  throughout  at  any  given  time,  that  is  practically,  as 
unelastic,  incompressible. 


Fig.  I.     The  external  and  the  middle  ear 

The  walls  of  the  tube  consist  of  hard,  unyielding  bone, 
except  in  two  places  where  the  bone  is  broken  through  and 
the  openings     closed     by     flexible     mem- 
The  tube  has  branes.     These   two   places   are   common- 

two  windows  to  ^Y  called  the  oval  and  the  round  windows, 
communicate  with  (The  fact  that  the  tube  communicates 
the  middle  ear  with  the  semicircular  canals  and  the  oth- 

er parts  of  the  labyrinth  can  here  be  neg- 
lected, since  all  these  communicating  cavities  are  also  enclos- 
ed in  bone,  not  possessing  any  windows.)  On  the  other  side 
of  these  windows  there  is  the  air  of  the  middle  ear.  Let  us 
now  consider  at  once  what  could  happen  to  the  fluid  in  the 
tube  if  rhythmical  changes  of  pressure  in  the  external  air  (a 
"tone")  caused,  through  the  tympanum,  like  changes  (of 
condensation  and  rarefaction)  in  the  air  of  the  middle  ear. 
Let  us  at  present,  however,  consider  this  under  the  imaginary 
assumption  of  no  chain  of  ossicles  existing  in  the  middle  ear. 
What  was  said  about  waves  in  the  fluid  of  the  tube  holds 
good  to  some  extent  also  for  the  air  in  the  middle  ear.  That  which 
occurs  there  is  the  same  as  that  which  occurs,  say,  in  a  bicycle 


4  UNIVERSITY  OF  MISSOURI  STUDIES 

pump,  that  is,  an  alternate  condensation  and  rarefaction  of  all  the 
particles  of  air  almost  simultaneously.  This  condensation  and 
rarefaction  always  acts  in  the  same  sense  (positive 
or  negative)  on  both  windows  of  the  tube.  According  to 
the  laws  of  hydrodynamics  no  motion  in  the  fluid  of  the  tube 
can  result  from  the  difference  in  size  of  the  two  windows.  It 
is  hardly  comprehensible,  therefore,  why  we  find  in  literature 
lengthy  discussions  of  the  question  whether  it  is  the  round 
or  the  oval  window  through  which  "the  tone  waves"  enter 
the  inner  ear.  They  do  not  enter  through  either  window 
since  they  do  not  occur  in  the  middle  ear,  the  volume  of  this 
cavity  being  too  small  to  contain  whole  tone  waves.  Only 
after  complete  destruction  of  the  tympanum  would  the  ques- 
tion as  to  the  manner  in  which  an  air  wave  strikes  the  tw^o 
windows  attain  practical  importance.  Under  normal  condi- 
tions we  must  regard  all  the  air  particles  in  the  middle  ear 
as  being,  at  any  time,  of  identical  density,  and,  thus,  as  unable 
to  produce  any  movement  in  the  inner  ear. 

If  there  were  no  ossicles,  the  fluid   in  the  tube  would 
remain  practically  motionless.     But  to  the  membrane  of  the 
oval  window  is  attached  the  plate  of  the 
Disturbances  stirrup  which  has  a  somewhat  ^id  con- 

within  the  tube        nection  with  the  tympanum.     The  result 

,      ^  is  that  every  movement  of  the  tympanum 

motion  of  .        ,  -     , 

the  stirniD  ^^    accompanied    by    a    movement    of    the 

stirrup  in  the  same  (positive  or  negative) 
direction.  Whenever  the  tympanum  moves  inwards,  the  air 
in  the,  middle  ear  is,  of  course,  somewhat  condensed.  But 
this  condensation  or  rarefaction  has  no  relevant  influence  on 
the  fluid  in  the  tube,  as  before  mentioned.  The  alternate 
condensation  and  rarefaction  of  the  air  in  the  middle  ear,  re- 
sulting from  like  processes  in  the  external  auditory  passage, 
is  an  unavoidable,  but  functionally  negligible  by-product  of 
the  mechanical  process  in   question,  bearing  no  direct  rela- 


MECHANICS  OF  THE  INNER  EAR  5 

tion  to  the  function  of  the  tube.  It  is  the  movement  of  the 
stirrup  which  causes  the  disturbances  in  the  fluid  of  the  tube 
which  we  have  soon  to  study  in  detail.  And  this  motion  of 
the  stirrup  is  made  possible  only  through  the  mediation  of 
solid  bodies,  the  auditory  ossicles. 

The  bony  connection  between  the  stirrup  and  the  tym- 
panum  would    serve   its    purpose    of    causing    movements    in 
the  fluid  of  the  tube  whatever  might  be 
The  auditory  ^^^    special    structure    of   this    connecting 

ossicles  are  a  link.     As  a  matter  of  fact,  it  is  arranged 

system  of  levers  in  such  a  particular  manner  that  it  acts  as 
a  lever  (or  system  of  levers),  the  large 
arm,  so  to  speak,  being  attached  to  the  tympanum,  the  small 
arm  to  the  stirrup.  This  effect,  however,  is  produced  in  dif- 
ferent animals  in  different  ways.  In  birds,  for  example,  (Fig. 
2)  there  is  no  chain  of  three  little  bones,  but  only  a  single 
bone,  a  rod  bearing  an  oval  plate.  The  leverage  of  this  sim- 
ple connection  is  explained  by  the  fact  that  the  tympanum 
and  the  window  plate  are  not  in   parallel   planes.     The  far 


j'^ 


<]i 


Fig.  2.     Schematic  representation  of 
the  leverage  in  birds 


more  complicated  connection  by  means  of  three  links  of  a 
chain  of  bones  in  most  of  the  mammals  has  been  theoretically 
studied  by  various  investigators  and  found  to  result  in  a  sim- 
ilar, but  probably  more  delicately  adjustable  leverage  than 
the  simpler  arrangement  in  birds.    The  advantage  of  the  lev- 


UNIVERSITY  OF  MISSOURI  STUDIES 


erag"e  is  easily  understood.  To  cause  a  fluid  to  move  along 
a  narrow  tube  requires  a  considerable  force  because  of  the 
friction  resulting  from  the  narrowness  of  the  passage. 
The  extent  of  movement,  on  the  other  hand,  may  be 
of  any  minuteness,  the  nerve  ends  certainly  being  sensitive 
to  the  very  slightest  curving  of  their  tufts  of  hairs  of  which 
we  shall  have  to  speak  again.  It  is  of  advantage,  therefore,  to 
gain  force  at  the  expense  of  magnitude  of  displacement. 

Someone  might  here  raise  the  question :  Why  are  there 
two  windows  when  only  one  of  them  has  a  solid  connection 
with  the  tympanum?     The  answer  to  this 
Why  would  not        question   is   very   simple.      If   there   were 
one  window  ^ot   a   second  window,   the   stirrup   could 

be  sufficient?  not  move  at  all.     Imagine  a  bottle  filled 

with  water  up  to  the  stopper  and  the 
stopper  fitting  the  neck  most  accurately.  Would  it  be  possi- 
ble to  drive  the  stopper  farther  in?  The  water  being  incom- 
pressible, it  would  not  be  possible  for  a  moderate  force 
to  drive  a  perfectly  fitting  stopper  in  any  more  than  to  pull  it 
out.  The  second  window,  closed  by  a  flexible  membrane, 
is  therefore  necessary  if  the  movements  of  the  stirrup  and 
of  the  fluid  in  the  tube  are  to  take  place.  If  it  were  not  for 
movements  of  the  fluid,  the  round  window  would  be  super- 
fluous. It  is,  however,  not  an  essential  condition  that  the  sec- 
ond window  should  open  on  the  middle  ear  and  not  perhaps 
directly  on  the  external  air  space;  for  instance,  on  the  exter- 
nal auditory  passage,  or  anywhere  on  the  skull.  But  it  is 
an  essential  condition  that  the  one  window  containing 
the  stirrup  plate  open  on  a  drum  and  that  the  plate  be 
rigidly  connected  with  the  external  membrane  of  this 
drum.  Thus  every  condensation  or  rarefaction  of  the 
air  outside  the  drum  must  result  through  movements  of  the 
tympanum  in  like  condensations  or  rarefactions  inside  the 
drum ;  the  movements  of  the  tympanum  must  result  in  move- 


MECHANICS  OF  THE  INNER  EAR  7 

ments  of  the  stirrup,  and  consequently  in  movements  of  the 
fluid  in  the  tube.  If  the  tympanum  is  destroyed  to  such  an 
extent  that  the  middle  ear  can  no  longer  act  even  imperfectly 
as  a  drum,  movements  of  the  fluid  in  the  tube  must  be  dif- 
ficult to  produce.  The  organ  is  then  deprived  of  its  normal 
manner  of  functioning — a  defect  which  does  not  necessarily 
involve  total  deafness,  yet  certainly  a  great  impairment  of 
the  sense  of  hearing. 

We  naturally  do  not  wonder  at  the  fact  that  the  round 
window  is  arranged  in  the  simplest  way  possible,  that  is, 
opening  on  the  middle  ear  not  far  from  the  oval  window. 

Let  us  now  attempt  to  determine  what  movements  would 
occur  in  the  tube,  caused  by  movements  of  the  stirrup,  if 
this  tube  were  a  perfectly  plain  tube,  con- 
The  movement  taining  nothing  whatever  but  an  incom- 
of  the  fluid  in  pressible  fluid.     It  is  a  decided  advantage 

a  plain  tube  to  study  first  a  case  as  simple  as  can  be 

imagined.  We  are  sure  that,  thus,  the 
elementary  foundations  of  our  thought  will  be  clear  and  not 
confused  by  the  influence  of  a  complexity  of  conditions  and 
a   sum   of   powerful   prejudices   which    almost   inevitably   ac- 


Fig.  3.     Movement  of  fluid  in  a  plain  tube 


company  a  complexity  of  conditions.  Let  us  try  to  keep  clear 
of  such  influences.  In  figure  3  we  see  the  anatomical  facts 
of  our  imaginary  case  diagrammatically  represented :  a  long 
and  narrow  tube,  two  windows  at  one  end,  one  of  these  win- 
dows containing  the  stirrup,  the  other  end  of  the  tube  closed. 


8  UNIVERSITY  OF  MISSOURI  STUDIES 

The  question  is  this :  What  will  happen  to  the  particles  of  fluid 
in  the  tube  when  the  stirrup  moves  slig-htly  inwards  or  out- 
wards? This  is  a  problem'  which  can  be  answered  either 
on  the  basis  of  our  general  knowledge  of  similar  processes 
or  by  means  of  a  special  experiment.  Let  us  first  try  the 
former  way.  When  the  stirrup  is  pushed  inwards  and  the 
round  window  outwards,  the  liquid  near  the  windows  must 
certainly  move  in  the  direction  indicated  by  the  arrows  in  the 
figure.  O'f  course,  the  direction  of  the  movement  would  be 
the  opposite  if  the  movement  of  the  stirrup  changes  its  sign 
and  pulls  instead  of  pushes.  But  what  would  happen  in  the 
fluid  at  the  other  end  of  the  tube?  At  x  or  even  at  y?  The 
answer  to  the  question  is  simple :  Nothing  would  happen. 
No  movement  of  any  kind  could  possibly  occur  there,  since 
there  is  no  sufficient  cause  why  any  movement  should  occur. 
The  friction  of  the  fluid  against  the  walls  of  the  tube,  which  is 
quite  considerable  in  a  narrow  tube,  must  prevent  any  spread- 
ing of  the  disturbance  beyond  a  very  near  limit.  That  is,  when- 
ever the  stirrup  moves  back  and  forth,  those  particles  of  the 
fluid  which  are  in  the  nearest  path  leading  from  the  oval 
to  the  round  window  must  move  accordingly.  All  the  rest 
of  the  fluid  remains  motionless. 

In  order  to  demonstrate  the  facts  just  mentioned  to  those 
finding  difficulty  in  understanding  that  from  the  general  laws 

of  hydromechanics  nothing  else  could  re- 
A  simple  suit  in  the  case  in  question  but  what  we 

experiment  have  just  described,  we  may  perform  the 

following  experiment.  A  box  containing 
white  clay  in  a  plastic  condition  has  two  circular  openinigs 
on  one  side,  not  far  from  each  other,  as  shown  by  figure  4 
in  cross-section.  We  now  press,  by  means  of  a  piston, 
into  one  of  the  openings.  A,  a  small  quantity  of  colored  clay, 
then  a  small  quantity  of  white  clay,  and  again  colored  clay 
until  the  latter  becomes  visible  on  the  outside  of  the   box 


MECHANICS  OF  THE  INNER  EAR  9 

at  the  other  opening,  B.  In  our  figure  we  see  at  a  and  b 
the  colored  clay  pressed  in  first.  The  part  protruding  be- 
yond the  outside  of  the  box  is  cut  away.  At  c  we  see  the 
white  clay  pressed  in  afterwards,  and  at  f  the  advance  guard, 
so  to  speak,  of  the  colored  clay  pressed  in  last.  Wha;t  has 
happened  within  the  box  is  obviously  this.  The  colored  clay 
pressed  in  first,  collects  inside  the  box  near  A  in  the  direc- 
tion of  B.     A  corresponding  amount  of  the  white  clay  with 


An    experiment   with 
plastic  clay 


which  the  box  was  filled  has  been  pushed  out  through  the 
opening  B.  The  white  clay  pressed  in  next  forces  up  the 
colored  clay  somewhat  as  a  mass  of  glass  is  blown  up  in  a 
glass  factory  to  form  a  bottle.  This  white  clay  is  forced  up  in 
turn  by  the  succeeding  colored  clay,  the  "bottle"  of  colored 
clay  increasing  its  dimensions  at  the  same  time.  During  this 
whole  time  and  afterwards  the  total  mass  moves  in  the  di- 
rection of   B.     However,   the   particles   of   clay  to   the   left. 


lO  UNIVERSITY  OF  MISSOURI  STUDIES 

nearer  the  openings,  move  much  more  quickly  than  those 
farther  to  the  right.  This  is  seen  from  the  fact  that  the 
left  wall  d  of  the  white  "bottle"  has  been  separated  entirely 
from  the  opening  A  and  is  just  getting  ready  to  disappear 
altogether  through  the  opening  B,  whereas  the  right  wall  e 
is  merely  beginning  to  sever  its  connection  with  A.  We 
have  here  a  simple  experimental  proof  for  the  statement  of 
the  preceding  paragraph  that  friction  prevents  the  spreading 
of  the  motion  beyond  narrow  limits,  causing  it  to  occur  as 
near  the  two  openings  as  possible.  Although  the  experiment 
in  this  form  does  not  show  it,  the  reader  hardly  doubts  that 
somewhat  farther  to  the  right,  say  six  inches  from  the  open- 
ings, no  motion  whatsoever  has  occurred  during  the  whole 
time.  The  quickest  motion,  of  course,  is  in  this  particular 
case  not  found  at  the  extreme  left,  at  g,  but  about  a  fourth 
of  an  inch  to  the  right,  since  the  friction  at  g  is  too  great. 
Without  entering  into  a  detailed  study  of  the  hydrodynamic 
problem  which  confronts  us  here,  in  which  friction  against 
the  walls,  internal  friction  in  the  fluid,  and  the  momentum 
of  the  fluid  play  their  roles,  let  it  be  sufficient  to  say  here 
that  the  motion  is  practically  limited  to  the  portion  of  the 
tube  near  the  windows  in  accordance  with  the  general  law 
of  nature  that  whatever  occurs,  occurs  with  the  least  pos- 
sible expenditure  of  energy.  Some  clay  is  pressed  in  at  A. 
The  same  quantity  has  to  pass  out  at  B.  This  can  be  made 
possible  by  many  kinds  of  displacement  of  the  particles  with- 
in the  box.  But  only  one  form  of  displacement  becomes 
actual,  the  one  that  requires  the  smallest  amount  of  work  to 
be  done  by  the  piston  at  A.  And  this  form  of  displacement 
consists  in  the  displacement  being  confined  to  the  neighbor- 
hood of  the  openings. 


MECHANICS  OF  THE  INNER  EAR  II 


Let  us  now  consider  another  imaginary  case  which  will 
contribute  towards  a  better  understanding"  of  the  processes 

actually   occurring   in    the    ear.      Suppose 
The  effect  of  a  ^  P^^^  ^^  ^^^  tube,  near  the  windows,  to 

rigid  partition  be   divided   by   an   inflexible  partition,   as 

within  the  tube       shown  in  figure  5.     It  is  self-evident  that 

in  this  case  every  movement  of  the  stir- ' 
rup  would  cause  the  particles  of  fluid  in  the  upper  and  lower 
division  of  the  tube  to  move  in  the  directions  of  the  arrows, 
parallel  to  the  partition;  and  the  particles  at  y,  at  the  end  of 
the  partition,  to  move  up  or  down.  But  the  fluid  farther  on 
in  the  undivided  tube  would  remain  motionless,  as  in  the 
former  case,  since  there  is  no  sufficient  cause  why  it  should 
move.     If  the  partition  extended  farther,  the  only  change  re- 


<^^ 


Fig.  5.     A  rigid  partition  in  the  tube 


3 


suiting  would  be  a  diminution  of  the  length  of  that  part  of 
the  tube  where  the  fluid  remains  permanently  motionless. 
If  the  partition  extended  to  x  (Fig.  5),  leaving  only  a  small 
opening  of  communication  between  the  upper  and  lower  division, 
all  the  fluid  within  the  tube  would  have  to  move  whenever 
the  stirrup  moves.  If  the  partition  extended  throughout  the 
tube,  leaving  no  communication  whatever  between  the  two 
divisions,  no  movement  of  the  fluid  could  then  take  place,  of 
course;  but  no  piston-like  movement  of  the  stirrup  could 
then  take  place  either. 


12  UNIVERSITY  OF  MISSOURI  STUDIES 

Let  US  now  imagine  a  third  case.  Suppose  a  partition 
to  divide  the  tube  lengthwise  into  two  divisions,  leaving, 
however,  a  small  opening  of  communica- 
The  effect  of  a  ^^°"  between  the  divisions  at  x.  Suppose 
flexible,  but  inelas-  further  this  partition  to  be  neither  per- 
tic  partition  with-  fectly  rigid  like  a  wall  of  hard  bone  nor 
in  the  tube  as  readily  yielding  and  in  turn  contract- 

ing as  a  thin  rubber  membrane,  but  to  be 
of  the  physical  nature  of  a  soft  leather  strap  somewhat 
loosely  stretched  out  between  the  opposite  sides  of  the 
tube  to  which  it  is  assumed  to  be  well  attached.  To  have 
something  definite  in  mind,  let  the  reader  think,  for  compar- 
ison of  its  function,  of  a  leather-seated  chair.  If  you  press 
from  below,  the  seat  yields  and  bulges  upwards;  but 
soon  it  stops  in  spite  of  your  effort.  If  now  you  sit  down 
on  the  chair,  the  seat  bulges  downwards;  but  again,  it 
soon  stops — how  could  it  otherwise  be  used  for  the  support 
of  your  weight?  But  what  is  particularly  important  to  note 
here,  is  the  fact  that  the  leather  seat,  after  it  has  bulged 
either  way,  may  continue  to  remain  thus  until  some  ex- 
ternal force  acts  upon  it  again  from  the  other  side.  Now 
let  us  consider  the  movements  which  would  occur  in  the 
fluid  of  a  tube,  divided  into  two  divisions  by  a  partition 
of  the  nature  just  described.  If  the  partition  could  yield 
indefinitely,  the  case  would  obviously  be  practically  the  same 
as  the  first  one  we  studied — ^without  any  partition.  That 
is,  the  fluid  would  move  near  the  two  windows  and  the  part 
of  the  partition  suspended  between  moving  masses  of  fluid 
would  move  with  the  fluid.  Farther  on  where  the  fluid  re- 
mains motionless  the  partition  would  remain  motionless  too. 
But  we  made  the  assumption  that  the  partition,  like  the  seat 
of  a  leather-seated  chair,  can  move  only  within  certain 
narrow  limits  up  and  down.  Now,  the  result  of  this  condi- 
tion will  be  this.    When  the  stirrup  begins  moving  inwards. 


MECHANICS  OF  THE  INNER  KAR  1 3 

the  part  of  the  partition  next  to  the  windows  must  follow 
the  movement  of  the  fluid  and  move  downwards.  But  soon 
it  has  reached  its  lower  limit.  Consequently  it  acts  now  as 
an  unyielding  partition,  the  effect  of  which  we  studied  in 
our  second  case  above.  The  fluid  just  above  and  below  this 
temporarily  unyielding  part  can  now  move  only  horizontally, 
but  the  particles  of  fluid  next  to  the  end  of  this  now  motionless 
piece  move  down  and  push  the  underlying  piece  of  the  par- 
tition down  until  it  has  reached  its  lower  limit.  And  so, 
gradually,  further  and  further  pieces  of  the  partition  come 


1  

2  , 

3  V 

4  V 

?  ^ 

6  /TTA 

7  '^^^^^^^^^^^^\,:^ 

8  / V 


Fig.  6.     The  partition  moves  within  an  upper 
and  a  lower  limit 


down  until  the  stirrup  stops  moving  inwards.  Figure  6 
shows  a  number  of  successive  stages  of  the  position  of  the 
partition  during  this  process.  The  vertical  scale  in  this  rep- 
resentation is,  of  course,  enormously  exaggerated  relative  to 
the  horizontal  scale.  But  at  once  after  stopping,  the  stirrup 
begins  to  move  in  the  opposite  direction.  At  once  the  par- 
ticles of  fluid  next  to  the  windows  (not  those  which  have 
moved  down  last)  move  upwards  and  take  the  corresponding 
part  of  the  partition  with  them  until  it  has  reached  its  upper 
limit.  Now  the  following  parts  come  up,  and  so  on  in  exactly 
the  same  way  as  before,  except  that  we  have  now  an  upward 
instead  of  a  downward  movement,  —  until  the  stirrup 
stops   moving   in   this   direction.      Let   us   remember   by   all 


14  UNIVERSITY  OF  MISSOURI  STUDIES 

means,  because  a  mistake  made  here  in  our  comprehension 
of  the  process  would  result  in  serious  errors  later,  that  the 
bulging  of  the  partition,  whether  up  or  down,  begins  in- 
evitably as  near  the  two  windows  as  possible,  and  that  fur- 
ther pieces  can  bulge  in  either  direction  only  under  the 
condition  that  all  the  pieces  nearer  the  windows  have  already 
reached  their  limit  in  that  same  direction. 

We  made  at  the  beginning     of  the  last  paragraph  the 
assumption  that  there  was  a  small  opening  between  the  two 
divisions  at  the  extreme  end  of  the  tube. 
A  safety  valve  Let  us  see  what  purpose  such  an  opening 

could  serve.  What  would  be  the  result 
of  an  extraordinarily  large  movement  of  the  stirrup,  so  large 
that  the  whole  length  of  the  partition  would  reach  its — upper 
or  lower — limit  of  position  before  the  stirrup  ceased  to  move 
in  the  same  direction?  The  result  would  be  either  an  enforc- 
ed stop  of  the  movement  of  the  stirrup  or,  if  the  external 
force  acting  on  the  tympanum  and  stirrup  was  too  violent,  a 
bursting  of  the  partition.  The  latter  disastrous  result,  how- 
ever, can  to  a  considerable  extent  be  guarded  against  by  the 
opening  in  question.  As  soon  as  the  total  length  of  the  par- 
tition has  bulged  the  fluid  will  begin  to  flow  through  this 
opening  from  one  division  of  the  tube  into  the  other,  until 
the  stirrup  stops  moving  in  the  same  direction.  Such  an 
opening  therefore  can  serve  as  a  kind  of  safety  valve  for  the 
protection  of  the  partition. 

After   having   studied   the  hydromechanical   function   of 
several  imaginary  tubes  wtith  divers  interior  equipments,  let 
us  now  turn  to  a  careful  survey  of  the 
The  anatomy  and    ^^^^^  which  the  anatomists  have  discover- 
physiology  of  ed  for  us  concerning  the  structure  of  the 

the  inner  ear  inner  ear.     Figure  7  shows  us  in  a  cross- 

section  all  the  important  details  which  have 
been     found     there     by     the     anatomists.       Hard     bone     pro- 


MECHANICS   OF   THE  INNER  EAR 


l6  UNIVERSITY  OF  MISSOURI  STUDIES 

trudes  from  diametrically  opposite  sides  of  the  bony  wall  of 
the  tube,  on  the  left  side  more  than  on  the  right.  But  the 
bone  does  not  protrude  far  enough  to  actually  cut  off  the 
lower  part  of  the  tube  from  the  upper.  While,  therefore,  we 
do  not  find  a  hard,  inflexible  partition,  we  find  indeed  some 
kind  of  a  partition  since  the  space  between  the  bony  protru- 
sions is  filled  with  a  delicate  structure  which  we  shall  have 
to  study  somewhat  in  detail.  This  structure,  which  we 
shall  always  refer  to  hereafter  as  "the  partition"  in  the 
inner  ear,  is  customarily  spoken  of  under  the  name  of  its 
discoverer  as  the  organ  of  Corti.  The  lower  part  of  this  par- 
tition has  been  shown  to  be  a  membrane,  generally  called  the 
basilar  membrane.  This  is  obviously  the  strongest  part  of 
the  partition,  capable  more  than  any  of  the  other  elements 
of  structure  to  resist  a  pressure  of  the  fluid  above  or  be- 
low. But  we  must  not  think  that  this  membrane  is  the  main 
part  of  the  partition  considering  its  volume.  It  is  rather 
small  in  bulk  compared  with  the  rest.  Above  the  membrane 
we  see  a  triangular  structure,  something  like  two  pillars  which 
have  fallen  towards  each  other.  This  structure  is  usually 
called  the  rods  of  Corti.  Its  mechanical  significance  becomes 
at  once  clear  to  us  when  we  see  at  its  sides  the  delicate  end 
organs  of  the  auditory  nerve  fibres.  These  end  organs  would 
obviously  be  crushed  by  the  push  of  the  fluid  which  occurs 
now  from  above,  now  from  below,  as  w^e  have  seen,  if  they 
were  not  braced  by  this  arch.  No  better  protection  could  be 
devised  than  this  triangular  structure  which  effectually  pre- 
serves the  natural  form  of  the  soft  tissues  as  the  skeleton 
does  in  the  total  animal  body,  without  interfering  with  a 
slight  bending  or  compression  of  the  tissues  of  the  partition. 
On  the  upper  side  of  the  partition,  opposite  the  basilar  mem- 
brane, we  notice  another  membrane,  but  much  more  delicate 
in  structure,  easily  torn  to  pieces  when  sections  are  made 
for  the  miscroscope.    This  membrane  touches  the  tufts  of  hairs 


MECHANICS  OF  THE  INNER  EAR  1 7 

which  are  the  extreme  peripheral  parts  of  the  sensory 
organs.  This  membrane,  however,  is  firmly  attached  to  the 
left  side  of  the  partition  only.  Its  right  end  is  free  or  seems 
to  be  almost  free.  The  kind  of  action  exerted  by  this  mem- 
brane upon  the  hair  tufts  can  only  be  guessed.  The  real  con- 
nections between,  and  the  physical  properties  of,  these  tissues 
are  not  well  enough  known.  We  may  perhaps  make  this 
action  a  little  clearer  by  assuming  that  the  upper  membrane, 
when  the  partition  bulges  upwards,  pulls  the  hairs  slight- 
ly, and  that  a  bulging  of  the  partition  downwards  means 
merely  a  relief  from  this  pull.  It  is  hardly  worth  while,  how- 
ever, to  enter  into  details  of  a  function  which  cannot  be  more 
than  hypothetical  since  there  are  no  data  upon  which  to 
base  any  more  definite  theory.  But  there  is  little  doubt,  that 
the  points  between  the  tufts  of  hairs  and  the  membrane  in 
question  are  to  be  regarded  as  in  the  strictest  sense  the  per- 
iphery of  the  sensory  apparatus  of  hearing.  And  we  shall 
scarcely  make  a  grave  mistake  in  assuming  that  a  double 
bulging,  back  and  forth  in  the  vertical  direction,  of  the 
partition  causes  a  single  shock  in  all  those  nerve  fibres  whose 
termini  are  located  in  this  part  of  the  partition,  and  that 
somewhere  in  the  neurons  a  new  process,  perhaps  a  kind  of 
chemical  process,  is  set  up  if  more  than  one  of  such  shocks 
are  received  in  quick  succession,  that  the  special  character 
of  this  new  process  is  dependent  on  the  frequency  with  which 
these  shocks  follow  each  other,  and  that  thus  we  perceive  a 
definite  tone,  occupying — according  to  the  frequency  of  shocks 
received — a  definite  point  in  the  total  series  of  sensations  of 
hearing. 


l8  UNIVERSITY  OF   MISSOURI  STUDIES 

In  the  preceding  paragraph  we  studied  briefly  the  ana- 
tomical elements  of  the  partition  in  their  mutual  relations.  We 
now  have  to  get  a  definite  idea  of  the 
The  Dhvsical  physical   properties  of   the  partition   as   a 

properties  of  whole  in   its  relation  to  the   surrounding 

the  partition  fluid.     These  properties  depend,  of  course, 

as  a  whole  on    the   properties   of   its   elements.      The 

partition  as  a  whole  can  certainly  not  be 
regarded  as  perfectly  rigid  and  unyielding  to  pressure.  It  con- 
sists of  tissues  too  soft  to  be  unyielding.  On  the  other  hand, 
we  cannot  possibly  assume  that  under  the  influence  of  pres- 
sure the  partition  could  bulge  to  any  large  extent,  for 
this  would  be  disastrous  to  the  delicate  end  organs  of  the 
nerve  fibres.  We  could  hardly  make  a  mistake,  then,  in  as- 
suming that  the  partition  can  yield,  but  only  withm  very  nar- 
row limits  up  as  well  as  down,  even  if  we  did  not  know 
anything  about  the  physical  properties  of  the  anatomical 
elements.  We  know,  however,  that  the  basilar  membrane  is 
a  comparatively  tough  structure,  probably  capable  of  consid- 
erable resistance.  We  are  justified,  then,  in  our  conviction 
that  the  whole  partition  bulges  in  response  to  pressure 
but  resists  such  pressure  as  soon  as  a  certain  rather  narrow 
limit  of  displacement  is  reached. 

Here,  however,  arises  another  question  of  the  greatest 
importance,  which,  unfortunately,  cannot  be  answered  with 
anything  approaching  accuracy.  This  is  the  question  as  to 
the  elasticity  of  the  partition.  Of  course,  all  the  elasticity 
the  partition  can  possibly  have  must  be  the  elasticity  of  the 
basilar  membrane.  The  basilar  membrane  is  the  only  one  of 
the  anatomical  elements  of  the  partition  which  might  have 
a  tendency  to  restore  spontaneously  the  whole  partition  to 
its  original  position  after  the  pressure  causing  the  displace- 
ment has  ceased  and  before  any  pressure  in  the  opposite 
direction  has  had  time  to  act  towards  this  result. 


MECHANICS  OF  THE  INNER  EAR  I9 

There  is  only  one  way  of  deciding  for  our  present  pur- 
pose the  question  as  to  the   elasticity  of  the  basilar   m&m- 

brane.  We  must  recall  our  knowledge 
Is  the  basilar  ^^  ^^^  elastic  properties  of  similar  mem- 

membrane  elastic?  branous  tissues  which  are  found  in  divers 

parts  of  the  human  body  and  elsewhere 
in  the  organic  world.  Now,  we  know  that  there  are  plenty 
of  membranes  in  the  body  which,  when  stretched  within 
certain  limits,  show  a  tendency  to  return  to  the  original 
shape.  But  they  never  remain  in  a  stretched  condition,  that 
is,  under  tension,  for  any  length  of  time.  Indeed,  they  would 
become  permanently  lengthened  if  they  remained  thus.  This 
is  the  consequence  of  a  universal  biological  law.  We  may, 
for  instance,  bend  a  sapling  and  expect  it  to  straighten  itself 
as  soon  as  we  let  it  go,  because  of  the  elasticity  of  the 
stretched  tissues  of  the  convex  side  and  the  compressed  tis- 
sues of  the  concave  side.  Btit  if  we  tie  it  in  this  bent  po- 
sition to  another  tree  and  return  after  a  year  and  cut 
the  tie,  we  find  that  it  has  adjusted  itself  to  the  position 
we  gave  it.  This  biological  fact  does  away  at  once  with  cer- 
tain theories  found  quite  frequently  in  physical  and  other 
textbooks,  which  speak  of  the  basilar  membrane  as  con- 
sisting of  a  great  number  of  stretched  strings,  comparable 
to  the  strings  in  a  piano.  These  theories  assert,  after  having 
introduced,  in  opposition  to  the  laws  of  biolog>%  the  idea 
of  a  permanent,  constant  tension  of  the  basilar  membrane, 
that  these  different  strings — as  in  a  piano — are  under  different 
tension  and  differently  weighted  and  that  they  serve  accord- 
ingly as  resonators,  responding  sympathetically  to  the  va- 
rious sounds  of  the  external  world.  However  pretty  this 
theory  of  "the  piano  in  the  ear"  may  appear,  authors  who 
expect  their  readers  to  accept  it  as  the  truth  should  first  of  all 
try  to  convince  them  of  the  possibility  of  living  animal  tissues 
retaining  their  tension  for  any  length  of  time  instead  of  ad- 


30  UNIVERSITY  OF  MISSOURI  STUDIES 

justing  themselves  to  the  permanent  stretching  and  thus  los- 
ing their  tension,  as  all  living  tissues  do.  We  shall  not,  of 
course,  entertain  for  a  moment  this  idea  of  the  basilar  mem- 
brane being  under  constant  tension,  since  our  aim  is  not 
unreality,  but  reality.  We  need  not,  therefore,  discuss  any 
further  the  assumption  of  the  presence  of  resonators  in  the 
inner  ear,  which  falls  with  the  above  rejected,  preposterous 
assumption  of  a  permanent  tension.  That  the  membrane  is 
capable  of  resistance,  as  it  probably  is,  means  something 
very  diflferent  from  the  assertion  that  it  is  under  constant 
tension,  which  is  biologically  impossible. 

The  actual  question  before  us  is  evidently  the  question 
as  to  the  elasticity  of  the  partition  as  a  whole.     Now,  we 
have    seen    that    the    only    element    of     it 
Is  the  Dartition         which,   according  to   its    structure,   may   be 
as  a  whole  regarded    as   elastic,   is   the   basilar   mem- 

elastic?  brane.    This  membrane,  however,  we  have 

found  to  be  quite  a  small  part  of  the  bulk 
of  the  partition.  If  the  partition  is  displaced  by  an  external 
force  and,  this  force  having  ceased,  is  caused  to  return  to 
its  original  place  by  the  tension  which  the  basilar  membrane 
has  just  suffered,  such  a  spontaneous  return  movement  must 
be  greatly  retarded  by  the  bulk  of  inelastic  tissues  of  the 
partition  which  the  membranous  part  of  it  has  to  drag  or 
shove  along  with  itself.  A  spontaneous  return  of  the  par- 
tition to  its  normal  position  must  be  therefore  very  slow 
when  compared  with  the  velocity  of  a  displacement  caused 
by  a  rather  powerful  external  influence  from  the  stirrup. 
Liet  us,  then,  keep  in  mind  that  with  respect  to  the  elastic 
properties  of  the  partition  there  are  only  two  alternatives : 
Either  the  basilar  membrane  is  practically  inelastic ;  then 
the  partition  as  a  whole  is  inelastic  and  cannot  sponta- 
neously return  to  its  original  position  after  having  been 
displaced.     Or  the  basilar  membrane  is  elastic ;  then  the  par- 


MECHANICS  OF  THE  INNER  EAR  21 

tition  can  spontaneously  return  after  having  been  displaced, 

but  wiith   a  velocity  that  is  only  very  small  compared  with 

the  velocity  of  its  displacement.       Of  the  two  alternatives 

the  latter  seems  to  be  the  more  probable. 

We  saw  on  a  previous  page,  in   our  second   imaginary 

case  of  a  partition,  that  the  fluid  moves  along  the  unyielding 

partition,    causing     friction    on    the    sur- 

_  .         f  ,,        faces  of  the  partition.  The  same  friction 

Protection  of  the 

surfaces  of  the         must  be  suffered  by  any  part  of  the  real 

partition  from  partition   as    soon   as   it   has   reached    its 

the  friction  upper  or  lower  limit  and  as  long  as  the 

of  the  fluid  stirrup    continues    to    move    in    the    same 

direction,  pushing  the  fluid  on  over  the 
initial  parts  of  the  partition.  If  we  had  to  design  an  ap- 
paratus to  function  thus,  would  we  not  see  that  the  sur- 
faces of  the  partition  were  sufficiently  protected  so  that  the 
rush  of  the  fluid  over  them  could  not  injure  them?  It  is 
interesting  to  raise  this  question  of  protection  with  respect 
to  the  actual  partition  in  the  tube.  If  we  look  above  at  fig- 
ure 7,  representing  a  cross-section  of  the  partition,  we 
notice  that  the  lower  surface  of  the  partition  is  well 
protected  from  injury  by  friction  of  the  fluid  by  a  part  of  its 
own  structure,  the  tough  basilar  membrane.  The  upper  sur- 
face, however,  with  its  delicate  sensory  cells  would  be  ex- 
posed to  injuries  by  friction  were  it  not  for  the  membrane  of 
Reissner  which  we  see  stretching  across  the  upper  division 
of  the  tube.  The  space  between  this  membrane  and  the 
partition  does  not  communicate  with  the  rest  of  the  upper 
division  or  with  the  lower  division.  It  would  therefore  be 
really  more  nearly  correct,  in  speaking  of  a  partition  divid- 
ing the  tube  into  two  divisions  which  communicate  through 
an  opening  at  the  extreme  end,  to  call  the  total  body  between 
the  membrane  of  Reissner  and  the  basilar  membrane  the 
partition.      No  movements   perpendicular   to   the   plane    of   the 


22  UNIVERSITY  OF  MISSOURI  STUDIES 

drawing  can  occur  in  the  fluid  below  the  Reissner  membrane. 
The  fluid  here  can  only  move  up  and  down,  pushing  or  pull- 
ing the  organ  of  Corti  into  its  limit  of  displacement.  No  fric- 
tion of  the  kind  above  referred  to,  which  might  do  injury  to 
the  delicate  tissues  of  the  organ  of  Corti,  can  therefore  take 
place,  and  the  problem  of  protection  from  friction  is  thus 
solved.  We  shall,  however,  in  order  to  make  our  language  as 
simple  as  possible,  restrict  the  term  partition  to  the  organ  of 
Corti,  neglecting  the  membrane  of  Reissner,  since  this  mem- 
brane, aside  from  the  important  protection  which  it  oflFers  to 
the  tissues  below,  does  not  seem  to  possess  any  function 
whatever. 

We  saw  on  a  previous  page  that  an  imaginary  partiticfn 
which  is  able  to  yield  to  the  pressure  of  the  fluid  only  within 
certain  limits  would  be  exposed  to  the  dan- 
The  safety  valve  ger  of  breaking  whenever  an  extraordina- 
rily powerful  external  force  tended  to 
cause  a  movement  of  the  stirrup  which  would  displace  more 
fluid  than  the  yielding  partition  could  make  room  for,  and 
that  this  danger  might  be  avoided  or  at  least  greatly  lessened 
by  an  opening  of  communication  between  the  two  divisions  at 
the  end  of  the  tube.  It  is  interesting  to  learn  from  the  re- 
searches of  the  anatomists  that  such  an  opening — a  safety 
valve,  as  we  may  call  it — actually  exists  at  the  extremity  of 
the  tube  of  the  cochlea. 

We  may  now,  after  making  ourselves  familiar  with  the 
structural  elements  of  the  inner  ear  and  their  physical  prop- 
erties, enter  into  a  discussion  of  the  actual  function  of  the 
organ. 


MECHANICS  OF  THE  INNER  EAR 


23 


We  have  thus  far  taken  into  consideration  only  a  single 
movement  of  the  stirrup,  in  either  direction.  We  must  now 
study  the  result  of  a  rh3rthmical  movement 
of  the  stirrup,  back  and  forth,  a  number  of 
times  during  a  certain  length  of  time.  In 
order  to  have  a  definite  case  before  our 
mind  we  will  suppose  the  stirrup  to  move 
back  and  forth  in  such  a  way  that  it  will 
describe  a  sine  curve  on  a  board  moving 
parallel  to  the  plane  of  the  paper.  In  fig- 
ure 8  is  represented  a  single  period  of  such  a  curve  in  a  hor- 
izontal position.  It  is  not  necessary,  however,  to  imagine  this 
definite  curve.  What  we  shall  have  to  say  will  apply  equally 
to  any  simple  periodic  movement,  whether  of  the  form  of  a 
sinusoid  or  of  a  combination  of  straight  lines  or  of  any  other 


Stimulations  of 
the  brain  resulting 
from  a  given 
rhythmical 
movement  of 
the  stirrup 


Fig.  8.     A  curve  representing  stirrup  movement 


curve  connecting  each  maximum  with  the  preceding  and  the 
following  minimum.  The  question  arises  then  by  what  means 
— computation,  simple  description  in  words,  or  otherwise — ^we 
can  obtain  a  clear  and  sufficiently  detailed  view  of  the  move- 
ments of  the  partition.  What  we  want  to  know  is  the  form 
of  motion  for  each  point  of  the  partition,  and  the  temporal 


24  UNIVERSITY  OF  MISSOURI  STUDIES 

relations  existing  between  all  the  several  movements.  Only 
thus  can  we  obtain  a  definite  view  concerning  the  nervous 
Stimulations  received  by  the  brain  as  the  result  of  a  given 
rhythmical  movement  of  the  stirrup.  In  order  to  find  the 
movements  of  the  partition  in  every  detail  we  might  try  com- 
putation since  this  is  the  method  wfhich  yields,  although  not 
always  the  clearest,  yet  in  general  the  most  accurate  results. 
Our  chief  task,  then,  would  be,  stated  again  as  definitely 
as  possible,  to  find  out  for  each  point  of  the  partition  which 

moves  at  all  the  exact  time  which  elapses 
Comoutation  of  from  a  jerk  down  to  a  jerk  up  and  from 
the  form  of  ^  j^^'k  up  to  a  jerk   down.      Figure    9    may 

motion  of  the  help  us  to  understand  the     conditions  of 

partition  computing  the  time  interval  in     question. 

Let  us  call  x  the  distance  of  any  point  of 
the  partition  from  the  point  of  x^  nearest  the  windows.  The 
length  of  the  part  of  the  partition  which  moves  in  response 
to  the  motion  of  the  stirrup  depends,  of  course,  on  the  ampli- 
tude of  the  movement  of  the  stirrup.  This  length  alone  is 
represented  in  the  figure.  What  is  farther  to  the  right  re- 
mains motionless.  Tlie  dotted  lines  above  and  below  rep- 
resent the  upper  and  lower  limit  of  each  moving  point  of 


--■^■1.--.-— ------- J- 


Fig.  9.     The  partition  in  the  tube  and  its 
limits  of  movement 

the  partition.  In  our  curve,  figure  8,  the  minimum,  at  A,  rep- 
resents the  position  of  the  stirrup  most  to  the  left,  the  max- 
imum, at  the  time  B,  the  position  of  the  stirrup  most  to  the 
right.  The  horizontal  line  represents,  of  course,  the  time.  To 
the  position  of  the  stirrup  at  A  corresponds  the  position  of  the 
partition   (in  figure  9)   in  its  upper  limit ;  to  the  position  of 


MECHANICS  OF  THE  INNER  EAR  25 

the  stirrup  at  B  the  position  of  the  partition  in  its  lower  limit. 
Let  us  now  find  out  when  any  arbitrary  point  jr,  is  jerked  up 
and  when  it  is  jerked  down,  measuring  the  time  from  A.  It 
is  obvious  that  the  amount  of  fluid  for  which  room  is  made 
by  the  piece  of  the  partition  from  x^  to  x^  moving  from  its 
upper  to  its  lower  limit  is  equal  to  the  amount  of  fluid  dis- 
placed by  the  stirrup  moving  inwards  through  the  distance 
measured  by  3;.  (For  convenience  we  place  the  zero  point 
of  the  system  of  coordinates  in  a  minimum  point  of  the  curve.) 
It  would  be  very  easy,  therefore,  to  find  the  equation  of  inter- 
dependence of  X  and  y,  if  the  following  conditions  were  ful- 
filled : 

1.     If  the  quantity  of  fluid  displaced  were  proportional  to 
the  horizontal  movement  of  the  stirrup. 

2.     If  the  partition  were  perfectly  in- 

Four  assumptions      ,     .  •       ^1    .    •  ^     re    • 

.-^  elastic;   that  is,  not  oirenng  any  resistance 

made  •  not  as  ^°  ^  displacement  until  either  of  the  limits 

hypotheses,  but  is  reached,  and  then  offering  absolute  re- 
fer the  sake  of  a     sistance. 

gradual  compre-  3.     if  the  distance  between  the  upper 

nension  ^^^  lower  limits  were  the     same  at  any 

point  of  the  partition. 

4.     If  the  width  of  the  partition  at  any  point  near  the 
windows  were  the  same  as  at  any  point  far  away  from  them. 

Let  us  temporarily  regard  these  conditions  as  fulfilled.   If 
they  are   fulfilled,  x  is  proportional  to  y.     That  is,   a  unit  of 
movement  of  the  stirrup     always  pushes 
Attempt  at  down    (or  raises,  as  the  case  may  be)   a 

computation  unit  of  the  partition  lengthwise.     Or,  ex- 

continued  pressed  in  a  formula: 

(I)  y=Cx 

where  C  is  a  constant  dependent  on  the  physical  properties 
of  the  organ. 


26  UNIVERSITY  OF  MISSOURI  STUDIES 

The  equation  of  the  curve  in  figure  8  is : 

(II)  y:=c(l  —  COS  2»7rWf)  ; 

that  is,  while   t  changes   from  zero  to    .X-  ,   y   changes    from 

n 

zero  through  c,  2c,  and  again  c,  back  to  zero.  We  now  sub- 
stitute Cx  for  y : 

c  (1 — cos  27rwO  =  Cx,  consequently: 

c 

(III)  cos  2-7r7i/ =:  1 —     X 

C 

This  formula  permits  us  to  calculate  t,  that  is,  the  exact 
time  when  any  point  of  the  partition  is  jerked  down.  But 
it  holds  good  only  for  the  time  from  A  to  B,  that  is,  while  the 
stirrup  moves  in  one  direction.  As  soon  as  the  stirrup  reverses 
its  movement  a  new  formula  has  to  be  applied,  since  the  move- 
ment of  the  partition  is  of  a  kind  which  is  mathematically 
called  a  discontinuous  function.  The  moment  when  the  stir- 
rup reverses  its  movement  and  the  farthest  point  of  the  par- 
tition has  been  jerked  down,  the  function  jumps,  so  to  speak, 
from  this  point  to  the  beginning  of  the  partition  and  the  first 
point,  nearest  the  windows,  is  jerked  up.  The  formula  to  be 
used  from  B  to  C  is  to  be  derived  by  substituting  {%c  —  y) 
for 3;  in  (I),  since  x  would  now  be  proportional  to  (2c  —  y). 
We  then  have  the  following  new  equations : 
(IV)  2c  —  3;  =  Cx. 

(II)  y  =  c  (1 — cos  27rnt),  consequently: 

r 

(V)  cos  2Trnt  =  X — 1 . 

c 

This  fcrmula   is    valid    from  B  to  C,    that  is    for  values  of 

t  varying    from   -L.     to     — »   while    the    partition    is    being 

2n  n 

jerked  upwards.  We  notice  that  the  only  difference  between 
the  right  side  of  (III)  and  the  right  side  of  (V)  is  the  sign. 
For  the  same  x  we  obtain  the  same  absolute  value  of  cos  2hmt, 
but  in  the  one  case  it  is  positive,  in  the  other  negative.  Now, 
it  is  easy  to  see  what  this  means  for  the  time  interval  between 
a  downward  and  an  upward  jerk  of  any  point  of  the  partition. 


MECHANICS  OF  THE  INNER  EAR  2*] 

Remembering  that  (III)  is  valid  for  jerking  down,  (V) 
for  jerking  up,  we  notice  that  the  arc  of  cos  ^Trnt  runs  through 
the  first  and  second  quadrant  while  the  partition  is  being 
jerked  down,  through  the  third  and  fourth  quadrant  while 
the  partition  is  being  jerked  up.  Therefore,  since  we  found 
that  the  time  of  jerking  down  of  a  definite  point  x.,  and  the 
time  of  jerking  up  of  the  same  point  are  subject  to  the  con- 
dition that  cos  ^TTfit  yields  the  same  absolute  value,  but  dififer- 
ing  in  sign,  the  time  of  jerking  up  must  be  found  in  a  quad- 
rant opposite  to  the  quadrant  wherein  the  time  of  jerking  down 
occurred,  never  in  an  adjoining  quadrant;  that  is,  if  the 
former  time  is  to  be  found  in  the  arc  ^Tfiit,  the  latter  must 
be  found  in  the  arc   ^.-jmit     +    — )»   since    the    addition    of 

-3^  to  t  means  the  addition  of  two  quadrants.  The  differ- 
2n 

ence  of  time,  therefore,  is   always     -L  .     In  other  words,  the 

2  n 

time  interval  from  a  jerk  down  to  a  jerk  up  and  from  a  jerk 
up  to  a  jerk  down  of  any  definite  point  is  with  this  particular 
curve  always  the  same,  being  exactly  one  half  of  the  whole 
period.  We  have  thus  found  by  computation  the  exact  move- 
ment of  the  partition  in  case  the  movement  of  the  stirrup 
is  of  the  form  of  a  sinusoid. 

We  have  seen  then  that,  provided  a  certain  set  of  condi- 
tions   (our  four    provisional  assumptions)    is  fulfilled,  and  pro- 
vided the  movement  of  the  stirrup  is  of  the 
Summary  of  form  of  a  simple  sine  (or  cosine,  as  this 

the  foregoing  means  the  same)    curve,     computation  of 

discussion  the  movement  of  the  partition  is  possible. 

But  computation  is  neither  particularly 
clear — at  least  those  who  are  not  professional  mathemati- 
cians will  think  so — nor  is  it  universally  applicable,  but  only 
in  a  few  cases  of  stirrup  movement,  the  above,  the  case  of 
straight  lines  connecting  the  maxima  and  minima,  and  a  very 
small  number  of  others. 


28  UNIVERSITY  OF  MISSOURI   STUDIES 

To  prove  that  computation  is  not  universally  applicable 
let  the  movement  of  the  stirrup  be  represented  by  the  function 

y  z=  c{2'  —  cos  2<^mt  —  cos  2'rrnt) 

and  let  m  be  equal  to  4  and  n  equal  to 
Computation  ^     ^^^^    g-      j^     ^^^^    ^^    ^  -^^    ^^^^^^ 

abandoned  .    „  i  •     n      tt  n 

musically  speaking).     JLven  m    a    case    like 

this,  by  no  means  far  fetched,  rather  the 
contrary,  computation  is  impossible  since  it  would  involve, 
as  the  mathematical  reader  may  easily  convince  himself,  the 
solution  of  an  equation  of  the  fifth  degree  in  order  to  find  the 
mutually  corresponding  values  of  y  and  t  for  the  maxima  and 
minima  of  the  curve.  Without  these  values  for  the  maxima 
and  minima,  which  are  the  points  of  discontinuity  of  the  func- 
tion representing  the  movement  of  the  partition,  we  could 
not  proceed  at  all.  It  is  out  of  the  question,  therefore,  to  ex- 
pect that  computation  pure  and  simple,  even  under  the  four 
assumptions  provisionally  made,  will  ever  give  us  a  satisfac- 
tory comprehension  of  the  function  of  the  inner  ear.  We 
must  look  for  other  means  in  order  to  obtain  our  end,  an 
insight  into  the  details  of  movement  of  the  partition. 

Let  us,  then,  try  to  represent  the  movement  of  the  par- 
tition in  the  above  case  as  well  as  in  others  graphically.     I 

,     ,       shall  offer  to  the  reader  two  methods  of 
Graphic  methods  -r-i.     r    ^     r  ^u 

of  determining         graphic  representation.     The  first  of  these 

the  exact  ^^  more  accurate  in  some  respects  than  the 

movement   of  second,  but  a  little  more  difficult  of  ap- 

the   partition  plication. 

The  vertical  axis  of  our  system  of  coordinates  in  figure 

10  may  represent  the  succession  of  points  of  the  partition,  be- 
ginning from  next  to  the  windows.     The 

First  graphic  horizontal  axis  may  represent  the  time.    I 

methc^  must  warn  the  reader     against     thinking 

that  the  figures  resulting  on  the  paper  are 

pictures  of  something  that  exists  in  the  ear  or  elsewhere.    The 


MECHANICS  OF  THE  INNER  EAR 


29 


figures  are  not  pictures  of  existing  things  but  merely  symbols 
of  a  function,  that  is,  of  the  time  when  any  point  of  the  par- 
tition is  jerked  up  or  down.  The  construction  of  the  figure 
is  based  on  the  following  considerations.  Let  us  mark  on  the 
paper  the  points  indicating  the  time  when  any  given  point 
of  the  partition  is  jerked.  When  we  shall  have  marked  a 
sufficient  number  of  such  points,  we  shall  draw  a  curve 
through  them.  But  how  do  we  find  the  points?  The  move- 
ment of  the  stirrup  is  represented  in  figure  8,  When  the  stir- 
rup   has   its    extreme   position   to   the   left    (according   to     Fig. 


Fig,  10.     Graph  of  the  times  when  each  point  of  the  partition  is  jerked  down 
(curves  of  odd  numbers)  and  up  (curves  of  even  numbers).    Compare  figure  8 

9)  and  just  begins  to  move  inwards,  we  mark  the  time  as 
zero  and  the  point  of  the  partition  which  is  jerked  down  also 
as  zero,  since  the  point  which  is  jerked  is  the  point  nearest 
the  windows.  In  figure  10  we  find  this  point  near  a.  As  the 
time  advances  (Fig.  8)  the  stirrup  moves  farther  and  farther 
inwards,  with  gradually  increasing  and  later  again  decreas- 
ing velocity.  A  further  point,  say  h,  in  figure  10  must  be 
located  somewhat  to  the  right  of  a  and  above  a,  since  a 
more  distant  point  of  the  partition  is  represented  by  a  higher 
position  of  the  mark  in  our  system  of  coordinates,  and  the 


30  UNIVERSITY  OF  MISSOURI  STUDIES 

fact  that  it  is  jerked  later  is  represented  by  a  position  farther 
to  the  right.  Now,  since  the  velocity  of  the  stirrup  increases 
as  shown  by  figure  8,  the  following  marks  have  to  be  placed 
higher  than  proportionate  to  their  advance  to  the  right.  That 
is,  points  marked  ofif  by  equal  steps  on  the  partition  are  now 
jerked  successively  in  briefer  time  intervals  than  before. 
Later  approaching  the  time  B  in  figure  8,  the  stirrup  moves 
again  more  slowl)^  and  the  marks  in  figure  10  advance  there- 
fore more  rapidly  towards  the  right,  as  seen  in  f  and  g.  If 
we  now  draw  a  complete  curve  through  the  marks  a,  h,  c,  d, 
^>  f>  i}  we  convince  ourselves  readily  that  the  new  curve  is 
the  same  curve  as  the  one  in  figure  8  from  A  to  B.  Of  course, 
if  we  have  not  chosen  the  same  vertical  and  horizontal  scales 
in  both  figures,  the  new  curve  must  appear  more  or  less  steep 
than  the  old  one.  But  the  selection  of  a  scale  for  a  graphic 
representation  is  entirely  a  matter  of  convenience.  Choosing 
identical  scales,  we  simply  have  to  transplant  the  first  half 
of  the  curve  in  figure  8  from  A  to  B  into  the  new  figure. 

But  now  the  stirrup  begins  to  move  in  the  opposite  di- 
rection, causing  the  partition  to  be  jerked  upwards  gradually. 
The  point  of  the  partition  nearest  the  windows  is  jerked  up 
first,  the  others  later  in  regular  order.  Now,  it  can  be  easily 
seen  where  we  have  to  place  the  further  marks  in  our  new  fig- 
ure, namely  /:,  /,  ;,  k,  J,  iii,  n.  We  find  them,  or  rather  immed- 
iately the  complete  curve  of  which  they  are  points,  by  simply 
turning  the  second  half  (B  to  C)  of  the  curve  in  figure  8  up- 
side down,  without,  however,  making  any  change  between 
right  and  left.  In  this  way  we  go  on,  simply  transplanting  the 
parts  of  the  stirrup  curve,  leaving  the  rising  ones  in  the  same 
position,  but  turning  the  falling  parts  upside  down. 

If  we  now  desire  to  find  out  for  any  point  of  the  par^ti- 
tion,  for  example,  for  .r  ,  the  exact  time  when  it  is  jerked 
down  and  when  it  is  jerked  up,  all  we  have  to  do  is  to  pass 
on   from  this   point    (Fig.   10)    to   the   right    (along  the   dotted 


MECHANICS  OF  THE  INNER  EAR  3I 

line),  since  this  direction,  according  to  definition,  represents 
the  time.  Our  first  crossing  of  a  curve  (in  e)  means  a  jerk 
down;  the  next  crossing  (in  /)  a  jerk  up;  and  so  forth.  That 
is,  the  odd  crossings  mean  each  a  jerk  down,  the  even  crossings 
each  a  jerk  up.  The  time  intervals  can  then  be  measured 
with  a  rule.  We  find  in  this  special  case  that  the  intervals  are 
all  equal.  We  have  thus  graphically  represented  the  exact 
movement  of  the  partition  in  a  case  where  the  movement  of 
the  stirrup  is  of  the  form  of  a  sinusoid.  The  same  graphic 
representation  is  applicable  to  any  given  curve,  however  com- 
plicated it  may  appear.  This  method  has  universal  validity. 
We  shall  soon  convince  ourselves  of  its  importance  for  the 
analysis  of  a  complicated  curve. 

We  can  easily  learn  from  the  graphic  representation  be- 
fore us  that  under  the  assumptions  provisionally  made  the 
stimulation  of    each     nerve     ending    can 

hardly  be  influenced  by  the  form  of  the 
What  movement        .  ^,    ^  •         i    xi.      ^i.- 

,  ^,       . .  stirrup  curve,  that  is,  whether  this  curve 

of  the  stirrup  f  >  ' 

produces  the  ^^  a  sinusoid,  or  made  up  of  straight  lines 

sensation  of  a  connecting  the  maxima  and  minima,  or  of 

single  tone  (free  any  other  shape,  provided  the  maxima  and 
from  overtones)  ?  minima  remain  unaltered.  Let  us  sup- 
pose that  each  "down"  means  a  shock  to 
the  nerve  end  and  that  the  "ups"  are  indifferent  as  to  ner- 
vous excitation.  We  see  immediately  (Fig.  10')  that  the  time 
interval  between  two  shocks  at  any  point  of  the  partition 
must  be  exactly  the  same,  since  each  down  curve  would  be 
exactly  like  any  other  down  curve,  whatever  the  shape  of 
the  up  curve.  (This  result  would  be  the  same  if  the  "ups" 
meant  excitation  of  the  nerve  end  and  the  "downs"  were  in- 
different.) That  is,  the  particular  shape  of  the  curve  rep- 
resenting the  movement  of  the  stirrup,  has  no  significance 
for  the  question  whether  a  single  tone  will  be  heard  or  not. 
If  all  the  down  curves  are  identical,  a  single  tone  only  is 


22  UNIVERSITY  OF  MISSOURI  STUDIES 

audible.  I  remind  the  reader,  however,  that  we  are  deriving 
this  conclusion  on  the  basis  of  our  provisional  assumptions, 
and  further,  that  we  are  speaking  here  of  movements  of  the 
stirrup,  not  of  rhythmical  pressure  changes  of  the  air  in  the 
external  ear  or  of  movements  of  a  tuning  fork  or  any  other 
vibrating  body.  In  discussing  later  the  effect  of  the  latter  • 
conditions  upon  the  stirrup,  we  shall  see  that  their  form  is 
not  necessarily  identical  with  the  form  of  the  stirrup  move- 
ment. 

As  yet,  we  have  studied  only  very  simple  movements  of 
the  stirrup.     Before  we  take  up  the  problem  of  how  the  inner 

ear  analyzes  more  complicated  move- 
The  physiological  ni^nts  of  the  stirrup,  we  ought  to  remem- 
condition  of  ber  that  we  have  not  yet     discussed  the 

tone  intensity  physiological  condition  of  tone  intensity. 

We  have  spoken  only  of  the  frequency 
with  which  shocks  are  received  by  the  nerve  ends.  But  the 
frequency  of  the  shocks  determines  only  the  attributes  of 
pitch  and  quality,  not  the  attribute  of  intensity  of  a  tone  sen- 
sation. Let  us  look  to  another  sense  organ,  the  olfactory 
organ,  for  a  suggestion.  On  what  physiological  condition 
does  the  intensity  of  an  odor  depend?  Although  we  have  no 
definite  knowledge  here  any  more  than  in  the  sense  of  hear- 
ing, we  have  reason  to  believe  that  the  intensity  of  an  odor 
depends,  or  may  depend,  on  two  conditions:  1.  The  num- 
ber of  nerve  ends  stimulated;  and  2.  the  concentration  of  the 
substance  which  stimulates  each  of  these  nerve  ends.  Ac- 
cepting this  suggestion  we  have  to  see  what  conditions  might 
determine  tone  intensity.  Only  these  two  can  come  up  for 
consideration,  so  far  as  I  can  see:  1.  The  number  of  nerve 
ends  which  receive  shocks  in  a  definite  frequency;  and  2<. 
the  suddenness,  the  impetuosity  with  which  each  nerve  end 
is  shaken  when  the  point  of  the  partition  in  which  it  is  lo- 
cated is  jerked  down.     Now,  the  second  of  these  two  conditions 


MECHANICS  OF  THE  INNER  EAR  33 

is  theoretically  almost  beyond  our  reach.  We  cannot,  in  the 
present  state  of  our  knowledge,  obtain  a  very  clear  idea  of 
differences  in  the  suddenness  with  which  the  nerve  ends 
might  be  shaken  in  different  cases.  It  will  be  best,  therefore, 
to  omit  this  factor  in  the  discussion  of  intensity  altogether, 
or  at  least  for  the  present,  rather  than  burden  our  theory 
with  arbitrary  hypotheses  the  usefulness  of  which  is  no  more 
probable  than  their  uselessne.ss.  At  present  we  shall  limit 
our  discussion  to  the  first  condition,  the  number  of  those  nerve 
ends  which  are  stimulated  with  equal  frequency. 

It  is  clear  that  the  number  of  nerve  ends  stimulated  de- 
pends in  some  way  on  the  length  of  that  part  of  the  partition 
which  is  jerked  up  and  down  in  a  certain 
.  frequency.    But  here  w*e  are  confronted  by 

^,     ^,  •.     ,  this  difficulty.     We  do  not  know  whether 

the  theoretical  ^ 

determination  of  ^^^  nerve  fibres  are  equally  distributed 
tone  intensity.  along  the  partition.     It  might  be  the  case 

Fifth  provisional  that  on  a  certain  length  of  the  partition 
assumption  near  the  windows  a  greater     number  of 

nerve  ends  were  found  than  on  an  equal 
length  farther  away  from  the  windows ;  or  the  reverse.  In 
our  present  state  of  knowledge  this  difficulty  cannot  be  over- 
come. In  order  to  go  on  with  our  theory,  we  have  to  make 
an  assumption.  We  shall  make,  of  course,  the  simplest,  the 
least  arbitrary  assumption.  We  assume,  provisionally,  that 
equal  parts  of  the  partition  lengfthwise  contain  equal  num- 
bers of  nerve  ends.  If  it  should  be  found  that  the  theory 
agrees  with  the  facts  of  auditory  observation  more  closely  un- 
der another  assumption,  we  would  have  to  substitute  this  for 
the  one  now  made.  Of  course  a  definite  answer  given  to  the 
problem  by  the  anatomists  would  be  more  satisfactory. 


34  UNIVERSITY  OF  MISSOURI  STUDIES 

We  can  measure  the  length  of  that  part  of  the  partition 
which  is  jerked  up  and  down,  only  by  the  aid  of  our  knowledge 
(if  we  have  any)  of  the  movement  of  the 
Another  difficulty  stirrup.  Now,  the  reader  will  recall  among 
in  the  theoretical  our  provisional  assumptions  the  one  that 
determination  of  the  width  of  the  partition  at  any  point  near 
tone  intensity  the  windows  is  the  same  as  at  any  point 

far  away  from  them.  But  the  anatomists 
tell  us  that  this  assumption  is  incorrect ;  that  the  partition  is 
about  twelve  (or  more)  times  as  wide  at  the  end 
as  near  the  windows.  Nevertheless  we  shall  provisionally 
make  the  assumption  of  proportionality  between  any 
length  of  the  partition  being  jerked  up  and  down  and 
the  extent  of  the  movement  of  the  stirrup  which  causes 
the  movement  of  this  piece  of  the  partition,  in  order  to  under- 
stand first  a  simpler,  though  imaginary,  case  and  to  proceed 
gradually  to  a  comprehension  of  the  actual,  rather  compli- 
cated function  of  the  partition.  Let  us  be  aware,  however, 
that,  having  thus  simplified  the  actual  conditions,  we  cannot 
expect  to  find  a  perfect,  but  only  an  approximate  harmony 
between  the  results  of  a  theoretical  analysis  and  the  direct 
observations  of  an  actual  sound  analysis  by  the  ear.  We 
may  find,  indeed,  w^ith  respect  to  tone  intensity,  rather  se- 
rious disagreements  between  the  facts  and  the  theory.  But 
these  disagreements  will  disappear  as  soon  as  the  theory  takes 
account  of  what,  for  simplicity's  sake,  we  provisionally  neg- 
lect. 

Making  the  two  provisional  assumptions  just  mentioned, 
we  can  theoretically  measure  the  intensity  of  a  tone  sensa- 
tion by  the  total  length  of  that  part  of  the 
Tone  intensity  partition  the  nerve  ends  of  which  are  ex- 

in  our  graphic  cited  with  one  definite  frequency.     In  our 

representation  graphic    representation   (Fig.  10)  the    inten- 

sity can  then  be  measured  by  the  vertical 
distance  between  the  horizontal  coordinate  and  the  top  of  the 
curves  which  represent  the  down  and  up  jerks. 


MECHANICS   OF   THE  INNER  EAR 


35 


We  discussed  above  the  result  of  a  simple  back  and  forth 

movement   of  the   stirrup.     Let   us   now   do  the   same   with  a 

more  complicated     movement.     Figure  11 

represents     the     new     stirrup     movement 

which  we  are  going  to  study.     This  curve 

is  approximately  the  one  represented  by  the 

equation 

y=  (1  —  cos  ^2t)  -J-  (1  —  cos  2Tr3i) ; 

which  justifies  us  in  saying  that  it  represents  physically  the 

sum   of  two   tones   of  the  vibration   ratio   2> :  3:     Let   us   apply 


Analysis  of  the 
combination 
2  and  3 


Fig.  II.     1  he  combination  2  and  3.    First  characteristic  phase 


the  same  graphic  method  to  this  case.  We  have  first  to  trans- 
plant the  part  of  the  curve  from  the  first  minimum  to  the  fol- 
lowing maximum,  A  to  B,  into  figure  12'.  Now,  when  the  stirrup 
reverses  its  motion,  the  parts  of  the  partition  near  the  windows 
begin  to  be  jerked  up.  Therefore,  the  curve  from  the  maximum  B 
to  the  next  minimum  C  has  to  be  turned  upside  down  and  then 
transplanted.  The  following  part  of  the  curve,  from  C  to  D,  must 
be  transplanted  in  its  original  upright  position,  but  placed  on  the 


36 


UNIVERSITY  OK   MISSOURI  STUDIES 


horizontal  coordinate  of  the  new  figure,  whatever  its  elevation  in 
the  original  curve  may  be,  since  every  reversal  of  the  movement  of 
the  stirrup  causes  at  once  a  movement  of  the  parts  of  the  parti- 
tion next  to  the  windows  and  only  later  a  movement  of  the  follow- 
ing parts.  So  we  continue  transplanting  each  section  of  the 
curve,  alternately  upright  and  upside  down.    This  figure  (Fig.  12) 


Tone  I 


Tone  2 


Tone  3 


Figure  I2.  The  combination  2  and  3.  First  characteristic 
phase.     (A  is  identical  with  G.)    Compare  figure  11 


is  to  be  interpreted  in  the  same  way  as  figure  10.  The  distances 
from  x^  to  ;r„  x^  to  x^,  and  x^  to  x^  represent  three  pieces 
of  the  partition,  x^  being  next  to  the  windows.  During  the 
unit  of  time,  w^hich  is  here  the  period  from  A  to  G,  all  the 
nerve  ends  located  between  x^  and  x  receive,  as  is  easily 
seen,  three  shocks,  counting  the  number  of  shocks  received 
by  the  number  of  downs  (or  ups,  since  this  distinction  between 
the  physiologically  effective  and  ineffective  direction  of  jerk- 
ing is  arbitrary,  for  want  of  better  know'ledge  as  to  the  man- 
ner of  excitation  of  the  nerve  ends).  All  the  nerve  ends  be- 
tween x^  and  x^  receive,  as  the  figure  shows,  counting  from 
left  to  right,  two  shocks  in  the  unit  of  time.     And  all  the 


MECHANICS   OF  THE  INNER  EAR  37 

nerve  ends  between  x^  and  x^  receive  one  shock.  The  nerve 
ends  located  farther  towards  the  apex  of  the  cochlea  do  not 
receive  any  stimulation  and  do  not,  therefore,  concern  us.  Hov/ 
many  tones  should  we  expect  then  to  hear  in  this  case?  The 
answer  is  as  easy  as  simple :  Three  different  tones,  since  shocks 
of  three  different  frequencies  are  received  by  the  several  nerve 
ends.  And  the  musical  relationship,  the  pitch,  as  we  say,  of 
these  tones  is  determined  by  the  relative  frequencies  found, 
which  are  3  and  2  and  1.  The  relative  intensity  of  these  tones 
is  to  be  measured,  in  accordance  with  our  remarks  in  the  pre- 
ceding paragraph,  by  the  relative  lengths  x„  x^,  x^  x^,  and 
x^  Xy 

A  movement  of  the  stirrup,   not  probably  exactly  like, 

but  similar  to  the  one  just  discussed  could  be  produced  by 

sounding     simultaneously    with     approxi- 

^        .  mately  equal  intensities  two  tuning  forks 

Two  important  -^7        ,  .       ,     .,       . 

facts*     Sound  representmg  the  ratio  of  vibration     rates 

analysis  and  ^  •^-    ^^  ^^  ^^^^^  known  that  we  hear  in  such 

production  of  a  case  three  different  tones,  3  and  2,  which 

subjective  vvie  may  call  "objective"  or  primary  tones, 

difference  tones  ^^^  ^^  ^^^^y^  ^g  rn^y  call  a  "subjective"  or 
difference  tone.  Some  further  facts  con- 
cerning such  subjective  or  difference  tones  will  be  mentioned 
subsequently  for  those  readers  who  are  not  familiar  with  the 
conditions  under  which  they  make  their  appearance.  The 
appropriateness  of  calling  the  subjective  tones  in  question 
"difference  tones"  will  then  become  apparent.  The  fact  that 
our  theory  of  the  function  of  the  inner  ear  and  actual  obser- 
vation in  this  case  agree  so  nicely,  is  highly  satisfactory  to 
us  and  ought  to  encourage  us  to  proceed  further  in  applying 
the  theory  to  other  special  cases  of  movements  of  the  stirrup. 
Let  us  keep  in  mind  that  our  theory  thus  far  has  explained  in 
a  special  case  two  most  fundamental  observations:  1.  That 
our  organ  of  hearing  is  capable  of   analyzing  a   compound 


38  UNIVERSITY  OF  MISSOURI  STUDIES 

acoustic  process;  and  2.  that  it  has  the  power  of  producing 
on  its  own  account  subjective  tones  which  no  study  of  mere 
external  conditions  could  ever  have  revealed  to  us  as  a  natural 
consequence  of  the  physical  processes  we  call  tones. 

We  saw  in  the  preceding  paragraph  that  all  the  nerve 
ends  between  .r,,  and  .i\  received  three  shocks  in  the  unit  of 
time.  A  measurement  of  the  distances  in 
A  problem  for  the  figure,  however,  shows  that  the  time 
future  solution  intervals  between  these  three  shocks,  al- 
though approximately  the  same,  are  not 
exactly  alike  (and,  moreover,  there  are  differences  in  this  re- 
spect between  the  several  nerve  ends  all  of  which  receive  three 
stimulations).  Now,  it  is  probable  that  the  particular  nervous 
excitation  set  up  in  each  ganglion  cell  by  these  three  stimula- 
tions of  its  terminal  fibre  and  thence  carried  farther  to  the 
brain,  may  be  just  the  same  in  either  case,  whether  the  shocks 
are  received  in  an  exactly  regular  rhythm  or  in  a  slightly  irreg- 
ular succession.  It  will  be  one  of  the  problems  of  the  future 
to  decide  what  is  the  limit  of  irregularity  which  must  not  be 
overstepped  if  the  sensation  produced  is  to  be  the  same  as  that 
of  a  regular  series  of  shocks  of  the  same  frequency.  At  pres- 
ent we  have  hardly  any  certain  data  upon  which  to  found  a 
decision.  We  must  leave  this  problem  open  for  the  present. 
It  would  be  well,  however,  to  remember  that  the  above  graphic 
representation  of  the  movement  of  the  partition — ^for  simplic- 
ity's sake — is  based  on  a  number  of  assumptions,  and  that 
the  actual  movement  of  the  partition  is  doubtless  somewhat 
different  from  the  one  which  is  here  under  discussion,  and 
which  contains  probably  only  the  essential  features  of  the 
actual  movement,  not  all  its  minor  details.  It  is  entirely  pos- 
sible, under  these  circumstances,  that  the  irregularity  in  ques- 
tion is  in  reality  much  less  considerable  than  it  appears  to  us 
now,  and  what  seems  to  be  an  important  problem,  may  turn 
out  to  be  no  problem  at  all.    The  reason  we  have  for  believ- 


MECHANICS  OF  THE  INNER  EAR 


39 


ing  that  the  actual  irregularity  might  be  less  than  the  one 
found  here,  is  that  in  the  graphic  representation  we  have  as- 
sumed a  movement  made  up  of  absolutely  sudden,  unpre- 
pared jerks,  with  intervals  of  perfect  rest  between  them.  The 
real  movement  is  probably  a  more  gradual  change  from  rest 
to  motion  and  back  to  rest;  and  the  result  of  this  might 
very  well  be  an  equalization  of  the  time  intervals  preceding 
the  shocks  received  by  the  nerve  ends.  This,  however,  is  not 
offered  as  a  solution  of  the  problem,  but  merely  as  a  sugges- 
tion for  the  future  investigator  of  this  subject. 

Let    us    try    another    method    of    graphically    representing 
the    movement    of    the    partition     under    the     provisional    as- 
sumptions made.     This  method  has  a  cer- 
Second  method        ^^^"  disadvantage  as  compared     with    the 
of  graphic  method  used  above,  in  being  less  accurate 

representation  of  with  regard  to  the  time  intervals,  but,  on 
the  movement  of  the  other  hand,  the  advantage  of  a  greater 
the  partition  simplicity  for  the  constructor  as  well  as 

for  the  reader.  The  extension  of  the  par- 
tition from  the  windows  towards  the  apex  of  the  cochlea  is  here 
represented,  not — as  before — by  the  vertical,  but  by  the  hori- 
zontal extension  of  the  figure,  from  left  to  right.  Figure  13 
shows  the  method  as  applied  to  the  same  curve  (Fig.  11)  which 
we  have  just  discussed.  The  first  thing  we  have  to  do  is  to 
draw  in  the  given  curve  (Fig.  11)  at  equal  distances  so  many 
lines  parallel  to  the  horizontal  coordinate  that  each  of  the 
maxima  and  minima  can  be  regarded  as  lying  on  one  of  these 
parallels.  If  this  is  not  easily  done,  then  any  arbitrary  number 
of  parallels  may  be  drawn.  But  the  drawing  as  well  as 
the  interpretation  of  the  new  figure  requires  a  little  more  atten- 
tion in  this  case,  because  we  have  to  consider  fractions.  In  this 
figure  there  are  thirty  equidistant  lines  drawn  parallel  to  the 
horizontal  coordinate.  A  greater  accuracy  than  this  would 
be  entirely  out  of  place,  since  our  representation  in  any  case 


40 


UNIVERSITY  OF  MISSOURI  STUDIES 


is  merely  an  approximate  representation  of  the  actual  move- 
ment of  the  partition.  These  horizontal  parallels  are  auxiliary 
lines,  serving  the  purpose  of  a  measuring  scale.  The  second 
thing  we  have  to  do  is  to  draw  a  second,  independent,  system 
of  auxiliary  lines  enclosing  a  corresponding  number  of  spaces. 
These  lines  are  the  thirty-one  vertical  parallels  in  figure  13. 
The  horizontal  lines  here  indicate  for  the  times  A,  B,  C,  and 
so  forth,  the  positions  of  the  different  points  of  the  parti- 
tion at  the  upper  or  lower  limit  of  movement.     The  vertical 


B 
C 
D 

E 
F 
G 


3 

:> 

c 

:> 

^ 

■5 

c 

D 
^ 

i 

C 

:> 

5 

-■ 

"■ 

■  ■"■ 

"" 

"■ 

" 

" 

"' 

" 

-i 



_ 

.. 

■f 

"■ 

■" 

~ 

"■ 

"" 

" 

■■ 

" 

" 

"^ 

■" 

■i 



.  -i 

_ 

- 

— 

--f 



"■" 

\- 

—  ■ 

"  — 

— 

■  " 

-- 

•  — 

—  - 

-- 

•  — 

-- 

■  — 

-- 

— 

"" 

— 

— 

—  ■ 







~ 



•  — 

— 

~^ 

-_  — 

3  2  ^      _ 

Fig.  13.     Successive  positions  of  the  partition.     The  combination  2  and  3. 
First  characteristic  phase.     Compare  figure  11 


auxiliaries  serve  the  purpose  of  cutting  off  the  partition  a 
number  of  equal  sections  corresponding  to  the  number  of  parts 
into  which  we  divided  the  total  amplitude  of  the  given  curve 
representing  the  movement  of  the  stirrup.  To  the  right  of 
these  sections  which  move  are  to  be  imagined  the  parts  of  the 
partition  nearer  the  apex  which  do  not  move  at  all  in  this  spe- 
cial case  and  which  do  not,  for  this  reason,  concern  us  here. 
At  the  time  A,  all  the  moving  parts  of  the  partition  are  at 
their  upper  limits,  since  the  stirrup  has  at  this  time  its  extreme 
outward  position.     From  A  to  B,  the  stirrup  moves  through 


MECHANICS  OF  THE  INNER  EAR  y|  I 

thirty  units  inwards,  pushing  down  successively  all  the  thirty 
sections  of  the  initial  part  of  the  partition.  We  find,  therefore, 
in  figure  13  at  B  all  the  thirty  sections  at  their  lower  limits. 
From  B  to  C,  the  stirrup  makes  an  outward  movement  through 
nineteen  spaces.  The  result  is  an  upward  movement  of  an 
equal  number  of  sections  of  the  partition.  We  find,  therefore, 
at  C  the  first  nineteen  sections  of  the  partition  at  their  upper 
limits.  All  the  following  parts  of  the  partition  remain  ex- 
actly in  the  positions  at  which  they  were  at  the  time  B,  since — 
according  to  the  assumptions  under  which  we  are  working — 
no  force  whatsover  has  acted  upon  them.  That  is,  the  sec- 
tions twenty  to  thirty  are  still  at  the  lower  limits,  and  the 
further  parts  of  the  partition  in  their  normal  positions.  From 
C  to  D  the  stirrup  moves  inward  through  six  spaces,  as  seen 
in  figure  11.  It  causes  therefore  the  first  six  sections  of  the 
partition  to  be  jerked  down.  In  this  position  we  find  them 
in  figure  13  at  D.  All  the  rest  of  the  partition  remains  exactly 
as  it  was  at  C.  That  is,  the  next  thirteen  sections  are  still 
at  the  upper  limits  and  the  following  eleven  still  at  the  lower 
limits  where  we  found  them  at  B.  From  D  to  E,  the  stirrup 
makes  an  outward  movement  through  six  spaces,  causing  an 
equal  number  of  the  initial  sections  of  the  partition  to  be  jerked 
up.  We  therefore  find  in  the  figure  at  E  the  first  nineteen  sec- 
tions of  the  partition  at  the  upper  limits,  the  following  eleven 
at  the  lower  limits.  From  E  to  F,  the  stirrup  moves  inward 
again  through  nineteen  spaces,  causing  nineteen  sections  of 
the  partition  to  be  jerked  down.  We  find,  therefore,  in  the 
figure  at  F  all  the  thirty  moving  sections  of  the  partition  at 
the  lower  limits.  From  F  to  G,  the  stirrup  moves  outward 
through  thirty  spaces,  as  seen  in  figure  11.  This  causes  thirty 
sections  of  the  partition  to  be  jerked  up.  So  we  find  in  figure 
13  at  G  the  whole  initial  piece  of  the  partition  which  moves 
and  therefore  alone  concerns  us,  at  the  upper  limit.  The  stir- 
rup has  now  reached  the  very  position  from  which  it  started 


42  UNIVERSITY  OF  MISSOURI  STUDIES 

at  A ;  and  the  partition  has  the  same  position  which  it  had 
then.  We  have  thus  graphically  represented  the  characteristic 
positions  through  which  the  partition  passes  during  a  com- 
plete period  of  the  movement  in  question. 

The  graphic  representation,  of  course,  is  only  a  means 
to    an    end.       We   have    to    read    off    from    this    representation 

how  many  shocks  are  received  during  the 
How  to  read  off  period  by  the  nerve  ends  on  each  section 
the  tones  heard  of  the  partition.  This  is  easily  done.  Let 
and  their  us  again,  for  want  of  definite    knowledge, 

intensities  make  the  assumption  that  a  jerk  down  of 

the  partition  means  a  stimulation  of  the 
nerve  ends,  and  that  a  jerk  up  is  irrelevant.  We  then  sim- 
ply have  to  go  down  in  the  figure  from  the  top  to  the  bottom 
and  count  the  number  of  times  each  section  is  jerked  down. 
The  first  section  is  down  at  B,  up  again  at  C,  down  for  a  sec- 
ond time  at  D,  up  again  at  E,  down  for  a  third  time  at  F, 
and  up  again  at  G.  The  nerve  ends  on  this  section,  there- 
fore, receive  three  shocks  during  the  period.  We  find  the 
same  number  of  stimulations  on  the  following  five  sections. 
Let  us  now  inspect  the  seventh  section.  It  is  down  at  B,  up  at 
C  and  still  up  at  D  and  E.  It  is  down  for  a  second  time 
at  F  and  up  again  at  G.  That  is,  the  nerve  ends  on  this  sec- 
tion receive  two  shocks  during  the  period.  The  same  is  true 
for  the  following  twelve  sections.  Let  us  now  look  at  the 
twentieth  section  of  the  partition.  It  is  down  at  B,  still  down 
at  C,  D,  E,  and  F ;  up  again  at  G.  That  is.  the  nerve  ends  here 
receive  only  one  shock  during  the  period.  The  same  holds 
for  the  following  ten  sections.  We  see,  then,  that  three 
tones  must  be  simultaneously  heard,  which  we  may  call, 
according  to  the  relative  frequency  of  stimulation,  the  tones  3, 
2,  and  1.  The  relative  intensities  of  these  tones  may  be  re- 
garded— under  the  provisional  assumption  of  a  uniform  dis- 
tribution of  nerve  ends  lengthwise  over  the  partition — as  six. 


MECHANICS  OF  THE  INNER  EAR  43 

% 

thirteen,  and  eleven,  according  to  the  number  of  sections  which 
receive  the  greater  or  smaller  number  of  shocks. 

Let  us  now  apply  the  second  graphic  method  to  another 
given  movement  of  the  stirrup,  which  will  make  clear  to  us 
another  interesting  property  of  the  ear  with 
Difference  of  respect  to  the  manner  in  which  this   or- 

phase.     Charac-       S^^  analyzes  an  objective  sound.    The  curve 
teristic  curves  of    the    stirrup    (Fig.   14)    is    made    up    of 

of  a  tone  combi-     two  component  curves,  very  similar  to  the 
nation  curves  composing  the  last  curve  discussed. 

That  is,  each  of  the  two  components  is  ap- 
proximately a  sinusoid,  one  of  a  period  equal  to  two  thirds  of  the 
other's  period,  both  of  approximately  the  same  amplitude.  The 
resultant  curve  is  constructed  here  as  before  by  measuring 
and  adding  together  the  ordinate  values  of  the  components 
in  the  drawing.  The  difference  between  the  present  case  and 
the  last  case  discussed  is  a  difference  of  phase.  If  the  reader 
should  not  know  what  this  means,  it  can  be  easily  understood 
by  the  aid  of  figure  14.  We  find  there  two  sinusoids,  one  with 
two  and  one  with  three  maxima  within  the  same  period,  which 
accordingly  may  be  called  curve  two  and  curve  three.  Now 
imagine  curve  two  moved  slightly  to  the  right  until  the 
minima  at  the  extreme  right  and  also  the  minima  at  the  ex- 
treme left  coincide.  We  then  have  exactly  the  case  discussed 
above ;  that  is,  the  addition  of  the  two  curves  would  result 
in  a  compound  curve  as  represented  by  figure  11.  The  curves 
of  figure  11  and  of  figure  14  may  be  called  the  characteristic 
curves  of  the  ratio  2:3,  because  they  are  the  two  extreme  forms 
between  which  the  compound  curve  changes  as  the  result  of 
a  change  of  phase,  that  is,  of  a  lateral  movement  of  curve  two, 
while  curve  three  remains  stationar)^  Let  us  convince  our- 
selves here  that  there  are  no  more  than  two  characteristic 
compound  curves.  If  we  move  curve  two  again  slightly  to  the 
right   the  same  distance  as  before,  that  is,  one  twelfth  of  the 


44 


UNIVERSITY  OF  MISSOURI  STUDIES 


period,  we  obtain  a  compound  curve  as  shown  in  figure  16, 
which  is  exactly  like  figure  14  when  read  from  the  right  to  the 
left.  And  if  we  change  the  phase  again  in  the  same  manner, 
that  is,  move  curve  two  again  one-twelfth  of  the  period 
to  the  right,  we  obtain  a  compound  curve  as  shown 
in  figure  18,  which  is  exactly  like  figure  11  only  turned 
upside  down.  We  shall  demonstrate  in  the  succeeding 
paragraphs     that     it     is     entirely     irrelevant     with     respect 


/»\ 


Fig.  14.     The  combination  2  and  3.     Second  characteristic  phase 


to  our  theory  whether  we  read  a  curve  from  the  left 
or  from  the  right,  in  its  first  position  or  turned  upside 
down.  We  shall  demonstrate  thus  that  there  are  in- 
deed only  two  compound  curves,  no  more,  which  are  character- 
istic of  a  combination  of  two  sinusoids.  This  is  an  important 
fact  because  it  makes  much  simpler  and  easier  our  task  of 
comprehending  the  function  of  the  inner  ear. 


MECHANICS  OF   THE  INNER  EAR 


45 


Theory  applied 
to  second  charac- 
teristic curve  of 
combination 
2  and  3 


Let  us  apply,  then,  the  second  graphic  method  to  this 
second  characteristic  curve  of  the  combination  2  and  3.  We 
locate,  in  figure  14,  the  horizontal  coordi- 
nate so  that  the  absolute  minima  of  the 
compound  curve  are  to  be  found  thereon. 
We  then  draw  a  number  of  equidistant 
lines,  say  thirty,  parallel  to  the  horizontal 
coordinate.  To  avoid  making  the  figure 
obscure  I  have  indicated  of  these  parallels 
only  those  which  pass  approximately  through  the  maxima  and 
minima  of  the  curve.  We  further  draw  a  system  of  thirty-one 
equidistant  vertical  parallels  enclosing  a  series  of  thirty  equal 
spaces  which  represent  succeeding  pieces  of  the  partition.  In 
this  system  of  auxiliaries  we  represent  the  positions  of  the  par- 
tition at  the  time  A,  B,  C,  and  so  forth.  At  A  in  figure  15  we 
find  all  the  moving  sections  of  the  partition  at  their  upper  lim- 
its, since  the  stirrup  has  at  this  time,  as  figure  14  shows,  the 
most  outward  position,  the  external  air  pressure  and  accord- 
ingly the  density  of  the  air  in  the  middle  ear  being  lowest.  At 
B  we  find  all  the  thirty  initial  sections  of  the  partition  down, 


C-- 


— 

-- 

-- 

■- 

— 

- 

- 

— 

-- 

— 

-- 

•- 

- 

-- 

-■ 

— 

-- 

-- 

- 

-- 

- 

-- 

-- 

._ 

-- 

- 

— 

-- 

- 



1 

1:^ 
r 

— . 

- 

" 

" 

" 

" 

" 

" 

" 

"" 

~ 

" 

..._ 

■  — 

._ 

- 

"■ 

~ 

" 

~ 

-- 

" 

"■ 

~ 

"■ 

"" 

" 

--r  — 

— 

h- 

"h 

■- 

_ 

"- 

-■ 

■^ 

-- 

" 

1 

\- 

.  - 

"" 

"" 

-j-j 



-- 

— 

-■ 

-- 

•  — 

—  ■ 

— 

— 

-• 

-  — 

-\ 

— 

— 

-- 

■  — ■ 

~ 

~ 



■~ 

— 

^  ~ 

~- 

"~ 

~ " 

. — — 

Fig.  15.     Compare  figure  14 


46  UNIVERSITY  OF  MISSOURI  STUDIES 

since  from  A  to  B  the  stirrup  has  moved  through  thirty  units 
of  space  inwards.  At  C  we  find  the  twenty-four  initial  sections 
raised  again  since  the  stirrup  has  moved  outward  through 
twenty-four  spaces.  At  D  the  eleven  initial  sections  of  the  par- 
tition are  at  their  lower  limits  since  from  C  to  D  the  stirrup  has 
moved  through  eleven  spaces  in  an  inward  direction.  From  D 
to  E  the  stirrup  moves  outwards  through  three  spaces.  Ac- 
cordingly we  find  at  E  the  first  three  sections  of  the  partition 
raised  to  their  upper  limits.  From  E  to  F  the  stirrup  moves 
inwards  through  eleven  spaces.  Accordingly  eleven  sections  of 
the  partition  must  be  pushed  down  to  their  lower  limits.  We 
find  the  first  three  down  at  F.  The  following  sections  up  to 
the  twelfth  were  already  down  at  E.  In  order  to  represent 
eleven  sections  of  the  partition  as  just  pushed  down  we  have 
to  place  at  F  the  twelfth  and  the  following,  including  the  nine- 
teenth, sections  of  the  partition  at  their  lower  limits.  Then  the 
first  three  and  the  latter  eight  make  up  the  total  number  of 
eleven  sections  pushed  down.  From  F  to  G  the  stirrup  moves 
outwards  through  twenty-four  spaces.  Accordingly  all  sec- 
tions of  the  partition  are  raised  to  their  upper  limits  except 
those  from  the  nineteenth  to  the  twenty-fifth  which  were 
already  at  their  upper  limits  at  F  and  therefore  simply  stay 
there.  So  we  find  the  partition  at  the  time  G  in  exactly  the 
same  position  in  which  it  was  at  A ;  and  we  must  find  it  again 
in  the  same  position  since  now  another  period  of  stirrup  move- 
ment begins,  exactly  like  the  period  just  discussed.  We  now 
have  to  read  oflf  the  tones  heard  and  their  intensities  in  the 
same  manner  as  we  did  this  before.  The  result  is  that  we  must 
expect  to  hear  the  three  tones  3,  2,  and  1  in  the  relative  inten- 
sities three,  sixteen,  and  eleven. 


MECHANICS  OF  THE  INNER  EAR 


47 


Comparing  our  analysis  of  the  curve  in  figure  14  with  the 
former  result  obtained  from  figure  11,  we  observe    that    in 

spite  of  the  remarkable  difference  of  ap- 
Practical         -  pearance  of  these  curves  to  the  eye,  the 

irrelevance  tones   which   we   expect   to   hear   are   the 

of  phase  same.     This  is,  of  course,  of  the  greatest 

importance  in  musical  practice.  Imagine 
the  unsurmountable  difficulties  if  the  director  of  an  orchestra 
were  responsible  for  the  phase  in  which  the  several  tones  pro- 
duced by  the  members  of  the  orchestra  acted  upon  the  audi- 
tory organs  of  each  hearer  in  the  concert  hall.  But,  as  it  is, 
each  hearer  perceives  the  same  tones  whatever  the  phases  of 
the  objective  processes  in  the  air.  N'ow  those  who  believe 
in  the  existence  of  a  system  of  strings  like  "a  piano  in  the 
ear,"  have  laid  much  stress  on  this  fact  of  the  practical  irrel- 
evance of  phase,  and  some  have  even  gone  so  far  as  to  say  that 
it  compels  us  to  assume  sympathetic  resonance  to  be  the  me- 
chanical power  of  the  auditory  organ.  I  need  not  persuade 
the  reader,  however,  that  such  a  compulsion  does  not  exist. 
Some  have  gone  still  farther  and  asserted  that  phase  differ- 
ence has  never  and  under  no  circumstances  any  influence 
whatsoever  upon  the  auditory  perception.  Their  theory  of  the 
mechanics  of  the  inner  ear  may  lead  to  such  a  consequence,  to 
an  absolute  irrelevance  of  phase.  Experiment,  however,  has 
not  yet  proved  that  phase  difference  of  the  sinusoidal  compon- 
ents of  stirrup  movement  has  never  any  influence  of  any  kind 
upon  the  perception.  Our  theory  has  shown  us  the  practical  ir- 
relevance of  phase  differences  and,  at  the  same  time,  left  a  pos- 
sibility for  slight  influences  of  this  kind  upon  the  perception, 
resulting  in  a  change  of  the  relative  intensities  of  the  sev- 
eral tones  heard.  The  intensities  of  the  three  tones 
for  one  phase  we  found  to  be  six,  thirteen,  and  eleven ;  for  the 
other  phase  three,  sixteen,  and  eleven.  That  is  to  say,  we 
would  hear  in  the  second  case  the  same  tones,  but  their  relative 


48  UNIVERSITY  OF  MISSOURI  STUDIES 

intensities  would  not  be  exactly  the  same  as  those  in  the  first 
case.  That  is,  difference  of  phase  may  be  irrelevant,  but  it 
need  not  be  so.  Let  us  recall,  however,  that  our  representa- 
tion is  only  a  rather  remote  approximation  to  the  actual 
movements  of  the  partition,  so  that  actually  the  influence  of 
phase  upon  the  perception  may  be  other  than  it  here  appears 
to  be.  What  is  important  is  our  insight  into  the  possibility 
of  a  slight  influence  of  this  kind. 


Fig.  16.     Compare  figure  14 


I  promised  to  demonstrate  that  the  application  of  our  the- 
ory yields  the  same  result  if  we  read  the  curve  of  stirrup  move- 
ment from  the  right  to  the  left,  or  turn  it 
Theoretic  irrel-        upside   down.     The   former  case   is   illus- 
evance  of  the  trated  by  figure  16,  which  is  exactly  like 

sign  of  the  co-  figure  14  when  read  from  the  right  to  the 

ordmates  left.     Figure  17  shows  the  successive  posi- 

tions   of    the  partition.     At  B   the  twenty- 
four  initial  sections  are  down.     At  C  the  first  eleven  of  them 


MECHANICS  OF  THE  INNER  EAR 


49 


are  up  again.     At  D  three  are  down  again.     From  D  to  E  the 
stirrup    moves    through    eleven    units      of    space    outwards. 


A 

B 
C 

D 

E 

r 

0 

5                                 J 

I                                1 

9                    24>^                  30 

V 4 1 

G 

Fig.  17.     Compare  figure  16 

Therefore  at  E  the  first  nineteen  sections  are  up,  eight  of  them 
being  up  already  at  D.     From  E  to  F  the  stirrup  moves  in- 


Fig.  i8.     Compare  figure  11 

wards  through  a  little  more  than  twenty-four  units  of  space. 
Therefore  at  F  thirty  sections  are  down,  five  of  them  being 


50 


UNIVERSITY  OF  MISSOURI  STUDIES 


down  already  at  E.  At  G  (equal  to  A)  all  the  thirty  sections 
are  up  again.  The  tones  to  be  heard,  which  the  reader  after 
all  the  previous  practice  in  this  task  can  easily  read  off,  are 
3,  2,  and  1  with  the  relative  intensities  three,  sixteen,  and  eleven. 


19 


B 


— 


a 


Fig.  19.     Compare  figure  il 


As  expected,  this  result  agrees  perfectly  with  our  analysis  of 
the  curve  in  figure  14. 

Let  us  now  demonstrate  that  turning  the  curve  upside  down 
has  no  influence  on  the  theoretic  result.  Figure  18  is  exactly 
like  figure  11,  only  turned  upside  down.     In  figure  19  we  see 


■\ 

n 

\ 

'  1 

1     1 

Fig.  20.     The  combination  34  and  25 


MECHANICS  OF  THE  INNER  EAR 


51 


f --^ ° ■ . .  __  . 


40 


-J^\sF^t^ 


4J, 


.  —  u  J.  LI  —  «.-! 


JOl 


Fig.  21.     The  combination  24  and  25.     Compare  figure  20 


52  UNIVERSITY  OF  MISSOURI  STUDIES 

the  successive  positions  of  the  partition  corresponding-  to  this 
curve.  The  interpretation  of  the  figure  is  so  simple  that  the 
reader  will  easily  read  off,  without  any  aid,  what  tones  are  to 
be  heard;  namely  the  tones  3,  2,  and  1  with  the  relative  intensi- 
ties six,  thirteen,  and  eleven.  This  is  exactly  the  same  result 
as  that  of  our  analysis  of  the  curve  in  figure  11. 

The  interval  studied  above  is  in  musical  terminology  that 
of  a  fifth.     Let  us  now  study  an  interval  which  is  even  small- 
er than  a  semitone.     The    compound    curve 
in    figure    2*0    is    made    up    of    twenty-four 
ine  tone  com-  vibrations  originating  from  one  source  and 

,  rt-  twenty-five     from     another.       Figure     2il 

shows  the  successive  positions  of  the  parti- 
tion corresponding  thereto.  The  initial 
section  of  the  partition  moves  up  and  down  twenty-five  times 
during  the  period.  We  may,  therefore,  conclude  that 
the  nerve  ends  located  here  will  transmit  to  the  brain 
a  process  resulting  in  the  sensation  of  the  tone  25. 
In  order  to  discuss  this  matter  with  more  accuracy, 
I  have  not  relied  only  upon  the  draftsman's  skill  in  con- 
structing the  compound  curve,  but  computed  the  ordinate 
values  of  some  of  the  maxima  and  minima.  Such  a  compu- 
tation is  exceedingly  tiresome  work,  since  for  each  pair  of  val- 
ues in  the  table  it  is  necessary  to  compute  twenty  or  more 
values  in  order  to  select  from  them  what  appears  as  the  maxi- 
mum or  minimum.  But  the  accuracy  of  this  method  can  be 
carried  to  any  decimal  desired.  We  learn  from  the  table  of 
these  values  that  the  relative  intensity  (when  determined  in 
the  same  way  as  above)  of  the  tone  25  would  be  nine  (that  is, 
200—191). 


MECHANICS  OF  THE  INNER  EAR 


53 


Interval  24:25,  Equal  Amplitudes 


Min. 

Abscissa 

Ordinate 

Abscissa 
Difference 

Point 

Ordinate 
Difference 

0 

0 

73 

0 

400 

Max. 

73 

400 

73 

I 

400 

Min. 

147 

2 

74 

2 

388 

Max. 

1540 

246 

— 

21 

Min. 

1620 

167 

80 

22 

79 

Max. 

1685 

221 

65 

23 

54 

Min. 

1755 

191 

70 

24 

30 

Max. 

1800 

200 

45 

25 

9 

Min. 

184? 

191 

45 

26 

9 

Max. 

1915 

221 

70 

27 

30 

Min. 

19S0 

167 

65 

28 

54 

Max. 

2060 

246 

80 

29 

79 

Min . 

3453 

2 

— 

48 



Max. 

3527 

400 

74 

49 

388 

Min. 

3600 

0 

73 

50 

400 

If  we  regard — quite  arbitrarily — the  time  from  one  stimu- 
lation to  the  next  as  measurable  by  the  abscissa  differences 

of  the  succeeding  maxima,  we  observe  that 
Do  we  hear  this  difference  is  about  one  hundred  and 

the  tone  25?  forty-seven  at  the  beginning  of  the  period, 

that  it  decreases  very  slowly  and  is  about 
one  hundred  and  forty-five  at  the  maximum  twenty-three, 
about  one  hundred  and  fifteen  at  the  maximum  twenty-five, 
the  same  at  the  maximum  twenty-seven,  and  that  it  increases 
gradually  till  the  end  of  the  period.  One  twenty-fifth  of  the 
whole  period  is  one  hundred  and  forty-four.  This  is  the  average 
abscissa  difference,  on  which  the  pitch  of  the  tone  heard  depends, 
since  the  abscissa  difference  is  inversely  proportional  to  the  fre- 
quency of  stimulation.     But  the  actual  abscissa  differences,  as  we 


54  UNIVERSITY  OF   MISSOURI   STUDIES 

have  just  seen,  deviate  from  the  average,  particular!}'  in  the 
middle  of  the  period.  Now,  some  one  might  prefer  to 
conclude  that  we  ought  not  to  hear  the  tone  25  all  the 
time,  but  at  first  a  tone  somewhat  lower  than  this,  gradually 
rising  slightly  and  falling  again  in  pitch  towards  the  end  of  the 
period.  Whether  we  should  draw  this  conclusion  I  will  not 
attempt  to  decide.  Neither  do  I  care  to  express  a  definite 
opinion  as  to  what  we  actually  hear.  Let  the  reader  who 
wants  to  know  this  find  it  out  by  an  experiment  of  his  own. 
What  I  must  point  out,  however,  is  the  fact  that  the  time  inter- 
val between  two  maxima  is  not  necessarily  the  time  between 
two  stimulations.  In  a  provisional  way,  the  interval  be- 
t\veen  two  maxima  or  between  two  minima  or  between  two 
points  of  inflection  or  between  two  points  of  any  other  name 
and  definition  may  be  used  thus,  but  let  us  always  remember 
that  this  is  only  a  provisional,  an  artificially  simplified  method, 
which  can  scarcely  yield  more  than  a  rough  approximation  of 
what  actually  happens. 

Another  section  of  the  partition  moves  up  and  down  twen- 
t3'-four   times    during   the   period.      The   length   of   this    section, 
which  determines  the  relative  intensity  of 
Do  we  hear  ^^^  tone  heard,  is  derived  from  the  table 

the  tone  24?  as  being  twenty-one   (221  —  200).     If  Ave 

look  at  the  time  interval  between  the 
successive  maxima,  we  find  this  to  be  at  the  beginning  of 
the  period  one  hundred  and  forty-seven,  to  decrease  gradually 
to  one  hundred  and  forty-five  at  the  maximum  twenty-three, 
to  be  two  hundred  and  thirty  from  maximum  twenty-three  to 
maximum  tvventy-seven  (maximum  twenty-five  has  disappeared, 
as  seen  in  figure  21),  and  to  fall  again  to  one  hundred  and 
fort3"-five.  Here  again.  I  will  not  attempt  to  decide  what  we 
ought  to  expect  theoretically,  because  we  have  no  right  to 
deduce  anything  definite  from  a  theory  in  a  direction  in  v/hich 
this  theory  is  as  j^et  professedly  indefinite,  in  which  it  obvious- 


MECHANICS  OF  THE  INNER  EAR 


55 


ly  lacks  as  yet  all  details,  owing  to  the  deficiency  of  the  requisite 
experimental  data.  I  can  only  repeat  here  what  I  said  in  the 
preceding  paragraph. 

Before  we  continue  this  attempt  at  an  interpretation  of 
figure  21,  let  us  consider  an  imaginary  case  the  application 
of   which    to    our    figure   will    soon    make 
No  indiscrimi-  itself   clear.     Imagine   that   during  half  a 

nate  counting  of  second  a  nerve  end  receives  in  regular  in- 
stimuli  allowed  tervals  fifty  stimulations,  but  during  the 
following  half-second  no  stimulations  at 
all ;  then  again  for  half  a  second  fifty  stimulations  in  regular 
intervals,  and  again  for  half  a  second  none ;  and  so  on.  What 
could  we  hear  in  such  a  case,  but  a  tone  for  half  a  second, 
nothing  for  half  a  second,  a  tone  again  for  half  a  second,  noth- 
ing again  for  half  a  second,  and  so  on.  And  what  tone  would 
it  be?  Plainly  the  tone  which  we  ordinarily  call  lOO, 
because  the  frequency  with  which  fifty  stimuli  are  received  in 
half  a  second  is  the  same  as  that  with  which  one  hundred  are 
received  in  one  second.  I  need  not  waste  any  effort  in  trying 
to  prove  what  is  self  evident,  namely  that  it  would  be  absurd 
to  count  in  a  case  like  this  simply  the  number  of  stimuli  during 
any  whole  second  and  to  expect,  these  being  fifty,  that  we  should 
hear  the  tone  50.  And  yet  this  way  of  counting  has  been 
actually  proposed.  B'ut  this  proposition  may  well  be  ignored. 
Now  let  us  return  to  the  interpretation  of  figure  21.  The 
third  section  of  the  partition,  the  length  of  which  is  twenty- 
four  (191  —  i67),  receives  stimulations  in 
What  beats  approximately  equal  intervals   until  about  the 

do  we  hear?  miximum     twenty-three    when   there   is    no 

stimulus  at  all  until  about  the  maximum 
twenty-nine.  With  the  rough  approximation  here  possible  we 
may  say  that  there  is  no  stimulus  during  about  one-tenth  of  the 
period.  From  our  discussion  in  the  preceding  paragraph  it  fol- 
lows that  during  about  nine-tenths  of  the  period  we  should 


56  UNIVERSITY  OF  MISSOURI  STUDIES 

hear  a  tone  and  during  one-tenth  of  the  period  we  should 
hear  nothing  so  far  as  the  nerve  ends  of  this  section  are  con- 
cerned. The  pitch  of  the  tone  we  must  expect  to  lie  between 
the  tones  34  and  25,  acording  to  the  probable  frequency  with 
which  the  stimulations  are  received  during  that  part  of  the  period 
during  which  they  are  received. 

It  is  plain  that  the  fourth,  fifth  and  following  sections  of  the 
partition  must  move  up  and  down  very  much  the  same  as  the  third 
section  does,  with  this  difference  only, 
The  "mean"  tone  that  for  each  further  section  the  pause 
when  no  stimulations  at  all  are  received 
becomes  longer  and  longer.  The  total  sensation,  then,  which 
is  derived  from  the  sum  of  the  nerve  ends  of  the  third  and  the 
following  sections  must  be  a  tone  of  a  certain  intensity  at  a 
certain  time  when  all  these  sections  mediate  the  sensation, 
but  becoming  weaker  and  weaker  as  one  after  another  of  the 
sections  stops  moving  until  for  a  moment  it  ceases  alto- 
gether, then  appearing  again  and  increasing  up  to  its  former 
intensity.  And  so  on  again  and  again.  That  is  to  say,  we 
hear  this  tone  "beating."  And  since  its  pitch  lies  probably 
somewhere  between  24  and  25),  between  the  ''primary" 
tones  (perhaps  its  pitch  is  not  quite  constant  but  may 
vary  slightly  during  each  period),  I  propose  to  call  it 
the  "mean  tone"  (German:  Zwischenton) .  The  question 
whether  we  hear  such  a  mean  tone  I  do  not  care  to  answer 
here,  this  discussion  being  devoted  to  theory,  not  to  experi- 
mental research.  Let  the  reader  who  desires  make  observa- 
tions of  this  kind  himself. 

The  farthest  section  of  the  partition  set  in  motion  by  this 
movement  of  the  stirrup  moves  up  and  down  only  once  dur- 
ing the  period.     The  nerve   ends  located 
The  difference  here  receive  one  shock  during  each  period 

tone  and  convey  therefore  the  sensation  of  the 

tone  1,  the  difference  tone  of  this  case.  The 
intensity  of  the  difference  tone,  corresponding  to  the  length 
of  this  section  of  the  partition,  is  two. 


MECHANICS  OF  THE  INNER  EAR 


57 


It  is  not  impossible,  however,  it  is  even  probable,  also 
that  a  few  of  the  sections  just  preceding  this  last  convey  the 
sensation  of  this  difference  tone,  instead  of  that  of  the  mean 
tone.  The  last  section  which  may  convey  the  sensation  of 
the  mean  tone  moves  only  twice  up  and  down  during 
the  period,  in  quick  succession.  This  double  move- 
ment is  followed  by  a  long  pause  during  which  no  movement 
occurs.  Now,  experimental  research  of  recent  years  has  prov- 
ed that  two  shocks  received  by  the  auditory  nerve  ends  may 
be  sufficient  to  give  the  sensation  of  the  tone  corresponding 
to  the  frequency  with  which  the  two  shocks  are  received — 
but  only  within  the  middle  region  of  the  tonal  series.  To- 
wards either  end  of  this  series  four,  six,  and  even  more  shocks 
are  found  to  be  necessary  for  the  sensation  of  the  tone  cor- 
responding to  the  frequency  of  the  shocks.  What,  then,  will 
be  the  consequence  of  choosing  the  tones  24  and  25  somewhat 
higher?  The  section  of  the  partition  which  makes  the  two 
up  and  down  movements  in  quick  succession  can  no  longer 
convey  the  sensation  of  a  short  mean  tone.  If  there  is  only 
one  period  of  movement,  no  sensation  at  all  will  then  result. 
But  if  many  periods  succeed,  it  is  much  more  likely  that  the 
double  movement  of  the  partition  section  will  have  the  effect 
of  a  single  shock  than  no  effect  at  all;  and  the  repetition  of 
this  shock  in  each  succeeding  period  must  result  in  the  sen- 
sation of  the  tone  1,  the  difference  tone. 

If  the  tones  24  and  25  are  chosen  still  higher,  it  becomes 
improbable  that  even  three  shocks  received  by  the  nerve  ends 
in  quick  succession  between  two  long  pauses  can  give  the 
sensation  of  a  short  mean  tone.  In  this  case  it  is  highly  prob- 
able also  that  the  second  section  before  the  last  conveys  the 
sensation  of  the  difference  tone.  And  so  a  few  more  of  those 
more  distant  sections  may  convey  the  sensation  of  the  dif- 
ference tone  instead  of  the  mean  tone. 


58  UNIVERSITY  OF  MISSOURI  STUDIES 

If  the  difference  tone  results  exclusively  from  the  func- 
tion of  the  nerve  ends  located  on  the  last  moving-  section  of 
the  partition,  its  relative  intensity  is  two,  according  to  the 
above  table.  But  if  the  difference  tone  results  from  the  func- 
tion of  the  nerve  ends  of  further  sections,  its  relative  inten- 
sity must  be  higher  and  the  maximum  intensity  of  the  mean 
tone  correspondingly  lower.  That  is,  the  phenomenon  of  a 
beating  mean  tone  must  be  the  less  pronounced  the 
more  audible  the  difference  tone ;  and  the  difference  tone  of 
a  small  interval  like  the  one  in  question  must  be  the  more 
audible  the  higher  the  pair  of  primary  tones  in  the  tonal  series. 

Summarizing  now  our  interpretations  of  figure  21,  we 
must  say  that  so  far  as  the  meager  data  reach  from  which  we 
can  draw  theoretical  conclusions,  the  fol- 
The  combination  lowing  seems  likely  to  be  the  total  im- 
34  and  25;  pression    (listening  with   one   ear,   having 

summary  the  other  ear    plugged)  :      1.      A  tone  25 

of  the  constant,  but  comparatively  weak 
intensity  nine;  2.  a  tone  34  of  the  constant,  but  compar- 
atively weak  intensity  twenty-one;  3.  a  mean  tone  (perhaps 
slightly  varying  in  pitch  during  each  period)  of  an  intensity 
which  varies  once  during  each  period  from  zero  to  a  definite 
maximum  intensity  and  back  to  zero.  This  maximum  inten- 
sity may  be  (under  the  most  favorable  conditions)  as  high 
as  (relatively)  three  hundred  and  sixty-eight,  but  must  be 
much  less  if  the  primary  tones  are  above  the  middle  region 
of  the  tonal  series.  Its  being  less  means  that  the  "beats"  are 
less  pronounced;  4.  a  difference  tone  the  relative  inten- 
sity of  which  may  be  (under  the  most  unfavorable  condi- 
tions) as  low  as  two.  Its  intensity,  however,  may  be  greatly 
increased,  at  the  expense  of  the  maximum  intensity  of  the 
beating  mean  tone,  in  case  the  pitch  of  the  primary  tones  is 
raised. 


MECHANICS  OF   THE  INNER  EAR 


59 


Before  we  take  up  the  theoretical  discussion  of  further 
tone  combinations,  the  reader  ought  to  obtain  some  informa- 
tion concerning  the  difference  tones  which 
Laws  of  ^^  ^^^^   ^^   addition    to    the   "objective". 

difference  tones   in   the   several   combinations.       To 

tones  give    such    information    of   this    kind    as    is 

indispensable,  I  shall  state  here  the  laws 
of  these  phenomena  in  as  clear  and  comprehensible  a  manner 
as  possible.  These  laws  given  below  do  not  pretend  to  tell 
all  the  difference  tones  which  we  might  possibly  hear  in  every 
possible  combination  of  objective  tones.  Neither  do  they  tell 
the  relative  intensities  of  the  difference  tones,  although  this 
is  a  matter  of  no  small  importance.  Laws  of  difference  tones 
of  this  scientific  perfection  are  as  yet  not  known  and  may 
never  be  known.  The  laws  below  merely  tell  those  differ- 
ence tones  which  one  is  most  likely  to  hear  in  those  combi- 
nations which  correspond  to  relatively  simple  ratios  of  the 
vibration  rates  and  are  therefore  (musically  and  otherwise) 
particularly  interesting.     These  laws  are  the  following  four: 

In  case  the  ratio  of  the  vibration  rates   does  not  differ 
much  from  1:1,  let  us  say  11 :  12,  or  9911 :  9980,  a  single  dif- 
ference tone   is   audible,   whose  pitch  corre- 
First  law  of  sponds  to  the  pitch  of  a  tuning  fork  the 

difference  tones  vibration  rate  of  which  is  equal  to  the 
difference  of  the  vibration  rates  of  our 
case.  In  addition  to  the  difference  tone,  however,  beats  are 
usually  clearly  audible,  and  a  mean  tone  may  be  audible  too 
which  lies  between  the  two  primary  tones.  If  the  interval  is 
quite  small,  this  mean  tone  is  usually  more  pronounced  than 
either  of  the  primary  tones,  particularly  when  we  hear  with 
one  ear  only,  having  the  other  ear  plugged.  The  beats  just 
mentioned  seem  to  be  the  fluctuations  of  the  intensity  of  the 
mean  tone  rather  than  of  the  primary  tones,  if  we  use  one 
ear  only. 


6o 


UNIVERSITY  OF  MISSOURI  STUDIES 


Second  law  of 
difference  tones 


A  second  class  of  ratios  which  is  of  particular  interest, 
is  that  of  the  ratios  whose  numbers  differ  by  one.     In  each 
of  these  cases  the  difference  tone  1  is  audi- 
ble, but  often  quite  a  number  of  additional 
difference  tones  can  be  perceived.     If  the 
numbers   of   the   ratio   are    rather    small, 
as  in  the  case  of  5:4,  all  the  tones  from  the 
highest,  that  is,  5,  down  to  1  are  without  any  great  difficulty 
noticeable.     As  we  study  ratios  of  increasing  numbers,  the 
tones  following  directly  upon  1   (in  a  rising  direction)   seem 
to  have  a  tendency  to  drop  out.    And  if  we  go  on  in  the  same 


Objective  tones 

3.  2 

4j  3 

5,  4 

6,  5 

7,  6 

8,  7 

9,  8 
lo,  9 


Difference  tones  easily  audible 
I 

2,  I 

3,  2,  I 

4,  3,  ?»     I 
5;  4,  ?,     I 

6,  5;  ?,       I         ■ 

7,  6,  5,     ?,     I 
?.  I 


way,  we  soon  find  only  one  difference  tone  left,  the  tone  1.  We 
have  then  simply  reached  a  case  in  which  the  difference  tone 
is  determined  by  the  first  law  above.  The  accompanying  table 
represents  this  class  of  ratios  with  their    difference    tones. 

A  third  class  of  ratios  are  the  ratios  made  up  of  com- 
paratively small  numbers,  representing  intervals  less  than  an 


MECHANICS  OF  THE  INNER  EAR 


6i 


Third  law  of 
difference  tones 


octave.  In  these  cases  three  difference 
tones  are  often  easily  noticeable,  one  cor- 
responding to  the  direct  difference  of 
the  vibration  rates  {h  —  /) ;  one  correspond- 
ing to  the  difference  between  the  lat- 
ter number  (h  —  /)  and  the  vibration  rate/  of  the  lower 
primary  tone,  that  is,  (2/  —  h) ;  and  one  corresponding 
to  the  difference  between  the  just  mentioned  differences  {h  —  Z) 
and  (2/ — h),  that  is  (2/z — 3/).  It  is  to  be  noticed,  however,  that 
a  difference  tone  is  rarely  audible  which  corresponds  to 
a  difference  larger  than  the  subtrahend;  for  example,  the 
primary  tones  9  and  5  produce  the  difference  tones  4  and  1, 
but  not  3  =  4 — 1,  or  at  least  not  an  easily  noticeable  tone  3,  three 
being  larger  than  one.  The  following  table  contains  a  few  ex- 
amples of  this  class : 


Objective 

tones 

Difference  tones  easily  audible 

8, 

5 

2>, 

2,       I 

5, 

3 

3, 

I 

9, 

5 

4, 

I 

1, 

4 

3, 

I 

II, 

7 

4, 

3,     I 

The  fourth  class  are  the  ratios  made  up  of  comparatively 

small    numbers,   representing  intervals   larger    than   an    octave. 

The  first  fact  to  be  noticed  here  is  the  lack 

of  an    easily   observable    difference    tone 

corresponding    to     the     direct     difference 

of  the  two  vibration  rates.    Such  a  tone,  if 

audible,  would  lie  between    the    primary 

tones.    As  a  rule,  only  one  difference  tone  is  easily  noticeable 

in  these  cases,  which  can  be  found  according  to  the  following 


Fourth  law  of 
difference  tones 


62 


UNIVERSITY  OF   MISSOURI  STUDIES 


rule :  Find  the  smallest  difference  between  the  larger  num- 
ber of  the  ratio  and  any  multiple  of  the  smaller  number.  The 
table  contains  a  few  instances  of  this  class : 


Objective  tones 

Difference  tones  easily  audible 

II, 

4 

1=3X4—11 

12, 

5 

2=12—2x5 

9. 

4 

1=9—2x4 

II, 

3 

1=4X3—11 

5, 

2 

1=5—2X2 

8, 

3 

1=3x3—8 

Let  me  repeat  that  the  above  rules  do  not  pretend  to 
represent  scientific  laws  in  the  strict  sense  of  the  word.    They 

are  stated  here  chiefly  for  a  practical  pur- 
The  use  of  pose.    If  the  reader  who  is  unfamiliar  with 

such  laws  difference  tones  will  use  the  above  "laws" 

as  directions  for  observation  and  obtain 
a  first  hand  knowledge  of  the  phenomena  of  difference  tones, 
he  will  be  more  interested  in  the  theoretical  discussions  which 
are  to  follow,  and  able  to  decide  for  himself  in  what  di- 
rections the  mechanical  theory  is  yet  most  undeveloped  and 
most  wanting  in  details. 

Let  us  apply  our  theory  now  to  the  combination  of  two 
sinusoids  of  the  relative  periods  nine  and  four,  that  is,  of  the 

relative  frequencies  4  and  9'.  The  com- 
The  combination  pound  curve,  representing  the  function 
4  and  9  f(.v)  =  1.99  -f-  sin4.j»r  -f  sin9x 

is  shown  in  figure  22.  The  period  is  made 
to  begin  and  to  end  with  the  lowest  ordinate  value  of  the 
function,  zero,  because  this  has  certain  technical  advantages. 


MECHANICS  OF  THE  INNER  EAR 


63 


It  is,  of  course,  in  a  periodical  function,  entirely  irrelevant  for 
the  mechanical  theory  what  point  we  regard  as  the  beginning 
of  the  period.  The  accompanying  table  contains  the  pairs  of 
corresponding  coordinate  values  of  all  the  maxima  and  min- 
ima of  the  curve.  These  values  are  found  by  computing  a 
large  number  of  pairs    of    values    and    selecting    from    them 


Interval  4:9,  Equal  Amplitudes 


Ordinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Max, 

+  169 

119 

36S 

P 

338 

Min. 

—  16 

318 

183 

Q 

185 

Max. 

+  75 

471 

274 

R 

91 

Min. 

—  199 

696 

0 

A 

274 

Max. 

+  110 

929 

309 

B 

309 

Min. 

—   2 

1094 

197 

C 

112 

Max. 

+  142 

1275 

341 

D 

144 

Min. 

—  189 

1512 

10 

E 

331 

Max. 

+  42 

1724 

241 

F 

231 

Min 

—  42 

1876 

157 

G 

84 

Max. 

+  189 

2088 

388 

H 

231 

Min. 

—  142 

2325 

57 

I 

331 

Max. 

+   2 

2506 

201 

J 

144 

Min. 

—  no 

2671 

89 

K 

112 

Max. 

+  199 

2904 

398 

L 

309 

Min. 

-  75 

3129 

124 

M 

274 

Max. 

+  16 

3282 

215 

N 

91 

Min. 

—  169 

3481 

30 

0 

185 

Max. 

+  169 

3719 

368 

P 

338 

those  which  have  the  highest  and  lowest  ordinate  values. 
This  computation  is  a  very  slow  process,  but  has  no  limit  of 
accuracy.  Figure  23  shows  the  positions  of  the  partition  be- 
longing to  the  maxima  and  minima  of  figure  32.  We  see  that 
at    A    the   initial   forty   sections  of  the    partition   are   in  their 


64  UNIVERSITY  OF  MISSOURI  STUDIES 

upper  positions.  At  B,  the  first  thirty-one  of  them  are  at 
their  lower  limits.  At  C,  the  stirrup  has  caused  eleven 
sections  to  assume  their  upper  limits.  From  C  to  D, 
the  stirrup  moves  inwards  through  fourteen  units  of  space, 
pushing  dow^n  the  eleven  sections  which  were  up  at  C,  leaving 
the  following  twenty  unmoved  since  they  are  down  already, 
and  pushing  down  three  more,  so  that  now  the  first  thirty-four 


The  combination  4  and  9 


sections  of  the  partition  are  down,  six  further  sections  are  up, 
and  all  the  following  ones  are  in  their  normal  positions.  From 
D  to  E  the  stirrup  makes  an  outward  movement  through 
thirty-three  units  of  space,  moving  up  the  first  thirty-three 
sections  of  the  partition.  From  E  to  F,  the  stirrup  moves 
inwards  through  twenty-three  units  of  space;  and  so  on.  At 
S,  we  find  the  partition  in  the  same  position  as  at  A,  our 
starting  point;  then,  a  new  period  begins. 

Let  us  now  try  to  interpret  the  figure.  We  can  easily 
see  that  the  first  eight  sections  move  down  and  up  again  nine 
times  during  the  period.  This  would  mean 
Do  we  hear  9?  that  the  nerve  ends  located  on  this  section 
convey  to  our  mind  the  sensation  of  the 
tone  9  of  the  relative  intensity  eight.  The  ninth  section  of 
the  partition  moves  down  and  up  only  eight  times  during  the 
period ;  but  after  our  discussion  about  the  omission  of  stimuli 


MECHANICS  OF  THE  INNER  EAR  65 

it  is  clear  that  w,e  should  not  be  justified  in  concluding  that 
we  must  hear  the  tone  8.  This  tone  would  be  audible  only 
if  the  frequency  with  which  the  stimuli  occur  on  the  ninth 
section  was  less  than  the  frequency  on  the  first  eight  sections. 
However,  there  is  no  reason  why  we  should  regard  the  fre- 


quency as  different.  It  seems  most  probable,  then,  that 
the  nerve  ends  of  the  ninth  section  convey  to  us  the  sensation 
of  the  tone  9,  but  with  a  short  pause  (or  possibly,  because  of 
the  after-sensation,  a  diminution  of  intensity  only)  at  the 
moment  about  G,  when  no  stimulation  takes  place.  Our  total 
impression  of  the  tone  9  is,  of  course,  the  sum  of  the  sensa- 
tions conveyed  by  all  the  nine  initial  sections.  This  means 
that  the  tone  intensity  perceived  would,  on  the  whole,  be 
nine ;  but  that  for  one  moment  in  each  period  this  intensity  of 
the  tone  might  suddenly  be  slightly  decreased.     It  does  not 


66  UNIVERSITY  OF  MISSOURI  STUDIES 

seem  improbable — so  far  as  our  theoretical  data  permit  us  to 
draw  a  conclusion — that  such  a  sudden,  but  weak  decrease 
in  intensity  might  become  noticeable  as  a  kind  of  just  per- 
ceptible "beat."  I  leave  it  to  the  reader  to  decide  experi- 
mentally whether  the  tone  9  in  this  combination  appears 
slightly  "rough"  or  perfectly  "smooth." 

The  tenth  and  eleventh  sections  of  the  partition   move 
down  and   up   six  times   during  the   period.     But   we   must 

remember  here  from  our  previous  discus- 
Do  we  hear  6?      sion  that — in  order  to  conclude  as  to  the 

tones  to  be  heard — no  indiscriminate  count- 
ing is  permissible.  Mere  counting  of  stimuli  would  indicate 
the  tone  heard  only  in  case  it  seems  probable  that  these  stimuli 
occur  in  equal  or  approximately  equal  intervals.  Now,  a 
survey  of  figure  23  does  not  make  it  appear  probable  that  the 
stimuli  on  the  two  sections  in  question  occur  in  even  approx- 
imately equal  intervals.  The  partition  moves  down  at  F 
and  remains  in  the  lower  position  until  it  moves  up  at  I.  It 
moves  down  at  J  and  immediately,  at  K,  up  again.  Down 
at  L  and  up  at  M.  In  this  upper  position  it  remains  until 
P,  when  it  moves  down.  At  Q  it  is  up  again,  to  stay  in  the 
upper  position  until  B,  when  it  moves  down.  At  O  it  is  up 
again.  At  D  it  moves  down,  at  E  up,  and  at  F  down  again. 
Are  we  justified  in  concluding  that  the  nerve  ends  located  on 
these  two  sections  of  the  partition  must  convey  to  our  mind 
the  sensation  of  the  tone  6  of  the  intensity  two;  or  any  other 
definite  sensation?  I  do  not  know  how  to  answer  this  ques- 
tion. If  we  knew  the  time  intervals  between  the  successive 
stimuli  exactly,  we  might  attempt  to  decide  whether  one  or 
the  other  sensation  would  be  more  or  less  probable  in  this 
case.  But  we  know  that  figure  23i  is  only  an  approximate, 
not  an  exact  representation  of  the  actual  movement  of  the 
partition.  It  is  a  certain  comfort  in  this  dilemma  that  the  prac- 


MECHANICS  OF  THE  INNER  EAR  67 

tical  importance  of  a  decision  in  this  case  is  rather  small,  for 
the  reason  that,  whatever  sensation  these  two  sections  might 
produce,  it  would  be  a  sensation  of  the  relative  intensity  two 
only,  a  rather  weak  sensation  compared  with  the  tones  which 
appear  theoretically  certain. 

The  twelfth,  thirteenth,  and  fourteenth  sections  of    the 
partition  move  down  at  B,  a  second  time  at  F,  a  third  time 

at  J,  and  a  fourth  time  at  P.  These  sec- 
Do  we  hear  4?         tions,  therefore,  move  down  and  up  four 

times  during  the  period  in  approximately 
equal  intervals.  The  five  following  sections  of  the  partition 
move  down  at  B,  a  second  time  at  F,  a  third  time  at  L,  and 
a  fourth  time  at  P.  These  sections,  therefore,  move  down 
and  up  four  times  during  the  period  in  approximately  equal 
intervals.  The  four  sections  from  the  twentieth  to  the  twenty- 
third  move  down  at  B,  a  second  time  at  F,  a  third  time  at  L, 
and  a  fourth  time  at  P.  These  sections,  therefore,  move 
down  and  up  four  times  during  the  period  in  approximately 
equal  intervals.  The  following  four  sections  move  down  at  B, 
a  second  time  at  H,  a  third  time  at  L,  and  a  fourth  time  at 
P ;  that  is,  four  times  during  the  period  in  approximately  equal 
intervals.  The  four  sections  from  the  twenty-eighth  to  the 
thirty-first  move  down  at  B,  a  'second  time  at  H,  a  third  time 
at  L,  and  a  fourth  time  at  P;  again,  four  times  during  the 
period  in  approximately  equal  intervals.  The  thirty-second 
and  thirty-third  sections  move  down  at  D,  a  second  time  at 
H,  a  third  time  at  L,  and  a  fourth  time  at  P.  These  sections, 
therefore,  move  down  and  up  four  times  during  the  period  in 
approximately  equal  intervals.  It  follows  that  according  to 
our  theory  we  must  expect  to  hear  the  tone  4  of  a  relative  in- 
tensity twenty-two,  since  it  is  produced  by  all  the  sections 
from  the  twelfth  to  the  thirty-third. 


68  UNIVERSITY  OF  MISSOURI  STUDIES 

The  thirty-fourth  section  of  the  partition  moves  down  at 
D,  and  up  again  at  O ;  down  at  P,  and  up  again  at  S.  That 
is,  the  nerve  ends  of  this  section  receive 
Do  we  hear  any  two  stimuli  during  the  period.  We  may- 
difference  tones?  expect  to  hear,  therefore,  the  tone  2'  of  the 
relative  intensity  one.  The  three  follow- 
ing sections  of  the  partition  move  down  at  H  and  up  again 
at  O.  The  nerve  ends  on  these  sections  receive,  therefore, 
one  stimulus  during  the  period.  The  next  two  sections  move 
down  at  H  and  up  again  at  S.  The  nerve  ends  here  receive 
one  stimulus  during  the  period.  The  fortieth  section  moves 
down  at  L  and  up  again  at  S.  The  nerve  ends  here  receive 
one  stimulus  during  the  period.  We  must  hear,  then,  the 
tone  1  of  the  relative  intensity  six.  The  tones  2  (weak)  and  1 
(strong)  are  the  only  difference  tones  in  this  case  which  we 
can  derive  from  our  theory  with  some  degree  of  certainty. 

Summarizing  now  the  results  derived  from  our  represen- 
tation of  the  movements  of  the  partition  in  the  case  of  the 
ratio  4:9,  we  find  that  we  must  expect  to 
The  relative  hear  the  tones  9,  4,  2  and  1,  with  the  rel- 

intensities  ative  intensities  nine,  twenty-two,  one  and 

compared  six;    leaving   out    of   discussion    the    doubt- 

ful sensation  of  the  intensity  two  which 
may  be  conveyed  to  our  mind  by  the  tenth  and  eleventh  sec- 
tions. Now,  it  is  quite  natural  to  ask  the  question  whether 
we  hear  these  tones  with  just  these  relative  intensities.  Un- 
fortunately, no  exact  answer  to  this  question  is  possible, 
because  this  matter,  owing  to  technical  difficulties  and  other 
circumstances,  has  never  been  experimentally  subjected  to 
accurate  measurement.  It  is  known,  however — what  also  ap- 
pears in  the  above  statement  of  our  results — that  in  a  com- 
bination of  two  tones  the  higher  one  loses  in  intensity,  com- 
pared with  the  lower  one.  Yet  it  is  doubtful  if  this  loss  in 
intensity  is  so  great  as  the  number  nine  indicates,  compared 


MECHANICS  OF  THE  INNER  EAR  69 

with  twenty-two.  The  present  writer  at  least  is  inclined  to 
doubt  this.  He  believes  that  the  theory,  representing  only  an 
approximation  to  what  actually  happens  in  the  organ  of  hear- 
ing, exaggerates  the  degree  of  this  loss  of  intensity  on  the 
part  of  the  higher  tone.  He  is  also  inclined  to  believe  that 
the  theory  exaggerates  the  relative  intensity  of  the  difference 
tone  1,  which  ^^^s  found  to  be  six.  In  reality,  this  tone  seems 
to  be  somewhat  weaker  than  is  indicated  by  this  number. 

Let   us    remember,   now,    the     provisional      assumptions 
which  we  made  in  order  to  render  the  graphic  representation 

of  the  movement  of  the  partition  as  sim- 
The  third  and  P^^  ^^  possible.     We  may  raise  this  ques- 

fourth  provisional  tion  :  Is  not,  perhaps,  the  above  disagree- 
assumptions  ment  between  theory  and  experimental  ob- 

recalled  servation  a  result  of  one  or  more  of  these 

provisional  assumptions?  I  shall  demon- 
strate that  this  is  indeed  the  case.  Or,  more  exactly,  I  shall 
demonstrate  that,  if  we  omit  one  of  these  assumptions  and 
take  into  account  in  its  stead  the  actual  anatomical  conditions 
so  far  as  these  are  known,  we  change  the  results  of  the  theory 
in  such  a  direction  as  to  diminish  the  exaggerated  loss  of 
intensity  of  the  higher  primary  tone  and  also  the  exaggerated 
intensity  of  the  difference  tone. 

The  partition    was    provisionally    assumed    to    be  of    equal 
width  all  along  the  tube.    As  a  matter  of  fact,  its  width  near 

the  windows  is  only  one-twelfth  or  one- 
The  partition  is  tenth  (measurements  differ  somewhat)  of 
narrower  near  what  it  is  at  the  far  end  of  the  tube.    And 

the  windows  further,  it  is  to  be  noted  that  the  width 

of  the  partition  does  not  increase  uniformly 
along  the  tube,  like  the  area  between  the  dotted  lines  of  figure 
24,  but  that  it  increases  first  rather  rapidly,  later  more  slowly, 
like  the  area  between  the  curved  lines.  The  figure,  however, 
does  not  represent  the  true  relation  between  the  width  and 


fJO  UNIVERSITY  OF  MISSOURI  STUDIES 

the  length  of  the  partition.  The  partition  as  a  whole  is  much 
narrower  in  comparison  to  its  length  than  appears  in  the  fig'- 
ure.  Let  us  try,  then,  to  get  a  clear  conception  of  the  func- 
tional significance  of  these  facts.  It  is  of  no  particular  im- 
portance, in  this  connection,  whether  the  measurements  upon 


Fig.  24.     Shape  of  the  partition 

which  the  following  considerations  are  based  are  more  or  less 
incorrect,  as  they  probably  are ;  for  our  intention  is  merely  to 
get  an  idea  of  the  general  direction  in  which  the  actual  shape 
of  the  partition  changes  the  results  of  a  theory  having  pro- 
visionally assumed  that  the  partition  is  everywhere  of  equal 
width. 

When  the  partition  yields  in  either  direction,  up  or  down, 
its  former  place  is  taken  by  the  fluid  of  the  tube.  Let  us  call 
the  quantity  of  fluid  which  has  taken  po- 
A  unit  of  stirrup  sitions  formerly  occupied  by  the  partition 
movement  equals  "the  displaced  fluid."  Now,  it  is  plain  that 
a  unit  of  dis-  the   quantity  of  displaced   fluid    must    al- 

placed  fluid  ways  be  approximately  proportional  to  the 

distance  through  which  the  stirrup  has 
moved  since  its  last  reversal  of  movement.  If  the  partition 
were  equally  wide  everywhere,  then  any  section  of  equal 
length,  far  from  or  near  the  window's,  would  make  room,  in 
moving  from  one  limit  to  the  other,  to  the  same  quantity  of 
displaced  fluid  as  any  other  section.  And  then,  plainly,  the 
length  of  that  part  of  the  partition  which  is  caused  to  move 
from  one  limit  to  the  other  would  always  be  proportional  to 
that  part  of  the    stirrup    movement    which    caused    it    to    move. 


MECHANICS  OF  THE  INNER  EAR 


71 


This  is  the  effect  of  our  provisional  assumption.  But  if  the 
partition  tapers  as  it  does,  a  unit  of  displaced  fluid  (corre- 
sponding to  a  unit  of  stirrup  movement)  is  made  room  for  by 
sections  of  the  partition  of  very  unequal  length  according  as 
the  displaced  fluid  unit  is  located  nearer  or  farther  from  the 
windows.  Where  the  partition  is  narrow,  a  longer  section 
would  have  to  move  in  order  to  make  room  for  a  unit  of  dis- 
placed fluid.  Where  the  partition  is  wider,  a  shorter  section 
would  make  room  for  the  same  quantity  of  fluid. 

Since,  then,  tone  intensity  depends  on  the  length  of  the 
partition  section  which  is  jerked   up   and  down,   and   since  this 

length  is  not  proportional  to  the  given 
The  computation  value  of  the  stirrup  movement,  it  is  use- 
of  a  table  ful  to  have  a  table  showing  the  partition 

lengths  corresponding  to  various  stirrup 
movements  in  order  to  get  a  clear  idea  of  the  influence  of  the 
tapering  of  the  partition  upon  the  relative  tone  intensities. 
To  simplify  the  computation  of  such  a  table,  it  is  well  to 
restrict  it  to  a  short  distance  from  the  windows,  so  that  we 


iw 


y 


■>x 


Fig.  25.     The  partition  widens 

may  approximately  assume  the  partition  to  increase  uni- 
formly in  width  within  this  distance.  Let  us  call  w  the 
smallest  width  of  the  partition,  near  the  windows ;  let  us 
assume  that  a  distance  from  the  windows  equal  to  50w  the 
width  of  the  partition  is  Gze/,  and  let  us  assume  a  uniform  in- 
crease of  width.  Let  us  call  y  the  width  at  any  point  of  the 
partition   and  x  the   distance  of  this  point   from,  the  beginning 


72 


UNIVERSITY  OF  MISSOURI  STUDIES 


near    the   windows.       We    then    know   (Fig.  35)   that  the  ratio 

y    ^  is  equal  to  the  ratio  of  

X  50zf 

y — w 6'w — w I 

X  50W  10 

X  IOW-\-X 


JJ/=W+ 


10  10 


The  area  described  by  the  cross-section  of  the  partition  in 

being  jerked  from  one  limit  to  the  other  may  be  called  a  at 

the  point  where  the  width  of  the  partition 

-,,  is  smallest,  a   at  any  arbitrary  point  of  the 

X lie  Hxca 

described  by  a  partition.  These  areas,  let  us  assume,  are 
cross-section  of  geometrically  similar.  This  assumption 
the  partition  possesses  a  higher    degree  of    probability 

than  what  would  follow  for  the  areas  from 
the  third  provisional  assumption  made  above  for  the  sake  of 
simplicity.  It  then  follows  that  the  ratio  of  the  areas  is  equal 
to  the  ratio  of  the  squares  of  the  widths  of  the  partition  at  the 
same  points. 


a 

-yi 

a 

'Uf 

a= 

ay- 

w^ 

For  y  we  substitute  its  value  found  above  and  have  then  the 
equation : 

a{\0'w-\-xY 
«=  -^ — ,    , 

The  left  side  of  this  equation  is  a  measure  of  the  area 
described  by  the  cross-section  of  the  partition  at  the  point 
X,  in  being  jerked  from  one  limit  to  the  other.  The  right 
side  of  the  equation  contains  the  variable  x,  the  distance  of 
any  point  of  the  partition  from  its  beginning  near  the  windows, 


MECHANICS  OF  THE  INNER  EAR 


73 


and  the  two  constants  a  and  w.  The  former  of  these  con- 
stants is  the  area  described  by  the  initial  point  of  the  partition 
in  moving  from  one  limit  to  the  other,  of  whatever  form  this 
area  may  actually  be  found  to  be.  The  latter  is  the  width  of 
the  partition  at  the  initial  point. 

The  mathematical  reader  immediately  sees  that  that  quan- 
tity F  of  displaced  fluid  for  which  room  is  made  by  a  move- 
The  quantity  of  ment  of  any  given  section  of  the  partition 
fluid  for  which  is  determined  by  the  following  equation, 
room  is  made  which  can  be  easily  integrated. 


7  ^i 


Fz=  /    '  adx 
J  X, 

In  order  to  integrate  this  equation  we  have  to  express  « 
as  a  function  of  x.  This  has  been  done  above  under  the  tem- 
porary assumption  of  a  uniform  increase  of  width.  The  re- 
sult is  stated  in  the  equation  just  preceding  the  last.  We 
then  have 

/x^     a 
{lOwArxfdx^^ 
x^  lOOW"^ 

=      ^         (lOw+;tr,)3  _  Uqw^x:)^  \ , 

where  x^  is  the  farther,  ^^  the  nearer  of  the  two  points  enclosing 
whatever  section  of  the  partition  is  in  question. 

If  the  section  in  question  is  an  initial  section  of  the  par- 
tition, then  X ,  is  equal  to  zero,  and  the  quantity  of  displaced 
fluid  is 

F=  -^—  \(\ow^x:f—{\ow)A 

Let  us  regard  the  partition  as  consisting  of  sections  each  of 
the  length  of  w.  We  can  find,  then,  the  quantities  of  dis- 
placed fluid  for  which  room  is  made  by  the  first  section,  the 
first  two,  the  first  three,  the  first  four,  and  so  forth,  sections 


74  UNIVERSITY  OF  MISSOURI   STUDIES 

by  making  x^    successively  equal  to    zv,    to    2w,   to  Bw,    to  4ru/, 
and  SO  forth.     If  ;tr  =  nw,  we  have 


F  = 


^oozv" 


{iow-\-?izvy — (lowy  ■■ 


aw  [ ,      ,    .  1 

3100  L^       '         J 

Let   us    arbitrarily   regard  as    the  unit    of   displaced 

fluid.     We  could  then  easily  compute  a  table  which  contains 

the  number  of  fluid  units  displaced  by  the 

Two  tables  number  n  of  partition  units.     If  the  num- 

possible  ber  of  partition  units  is,  for  example,  three, 

the  quantity  of  displaced  fluid  is 
(13' — 10')   units  and  so  on. 

More  useful,  however,  is  a  table  which  progresses  in  a 
regular  series  of  units  of  fluid  and  tells  us — in  decimals — the 
lengths  of  the  initial  sections  which  make  room  for  these  quan- 
tities of  fluid ;  for  our  representation  of  the  movement  of  the 
partition  tells  us  the  quantities  of  displaced  fluid,  and  the  cor- 
responding section  lengths  are  to  be  found  in  order  to  obtain 
a  more  correct  idea  of  the  relative  tone  intensities.  In  order 
to  compute  such  a  table  it  is  advantageous  to  use  a  larger 
fluid  unit  than  the  above.  Let  us  determine  the  total  quan- 
tity of  fluid  for  which  room  is  made  by  the  partition  section 
from  .r  r=  0'  to  x  =  bOro,  that  is,  the  whole  part  of  the  partition 
near  the  windows  for  which  we  have  assumed  a  uniform  taper- 
ing or  change  of  width ;  and  let  us — arbitrarily — regard  one- 
fiftieth  of  this  quantity  as  the  fluid  unit. 


„      aw  F/       ,      N  1 

F  =  —    (lo-f  50)3—  103  = 
300  L  J 


2 1 5000  a  w 
300 


MECHANICS  OF  THE  INNER  EAR  75 

The  fluid  unit,  defined  as  one-fiftieth  of  the  above  quantity, 
is  therefore 


4300  a  w 

300 

Any  number  m  of  such  fluid  units  is  then 

4300  awm 
300 

We  derived  above  the  following  equation  between  fluid 
quantities  and  partition  lengths 


300  w  \_;-  -'     ^      '  \ 


In  this  equation  we  have  to  substitute  for  F  the  above 
expression  of  fluid  quantity  and  then  to  solve  the  equation 
for  x^. 

4'iooawm  a       \,  i       \^      /t^^.asI 

^^ =  (low  +  x^r  —  {lowf 

300  300w^  L^  J 


iow-\-x,  =  zvWiooo-{-42>OQm 


w 


x^  =  (Viooo+430o;«  —  lo) 


The  following  table  contains  the  corresponding  values  of 
m  and  x^,   measured  in  the  unit  of  length  w. 


76 


UNIVERSITY  OF   MISSOURI  STUDIES 


Let  US  see,  now,  how  we  must  use  the  table  of  fluid 
quantities  and  partition  lengths.     We  recall  that  any  unit  of 

stirrup  movement  causes  the  displacement 
The  use  of  of     a     unit     of     fluid.       What     we     have 

the  table  called  above  "the  relative  intensities  of  the 

tones  heard"  refers  directly  to  relative 
numbers  of  units  of  stirrup  movement ;  indirectly  also  to  rela- 
tive numbers  of  units  of  displaced  fluid,  since  it  is  highly  prob- 
able that  the  quantity  of  displaced  fluid  is  approximately  pro- 
portional to  the  extent  of  a  stirrup  movement.  What  we  want 


TABLE  OF  THE  RELATIONS  BETWEEN  FLUID  DISPLACEMENT  AND 
PARTITION  LENGTH 


m 

I 

X 

II 

X 

m 
21 

X 

m 
31 

X 

m 
41 

X 

7-43 

26.42 

35-03 

41.22 

46.18 

2 

11.25 

12 

27.47 

22 

35-73 

32 

41-75 

42 

46.62 

3 

14.04 

13 

28  46 

23 

36.40 

33 

42.28 

43 

47.07 

4 

16.30 

14 

29.41 

24 

37-05 

34 

42.80 

44 

47-50 

S 

18.23 

15 

30.31 

25 

37-70 

35 

43-31 

45 

47-94 

6 

19.92 

16 

31-17 

26 

38.32 

36 

43-81 

46 

48.36 

7 

21-45 

17 

32.00 

27 

38.92 

37 

44-30 

47 

48.78 

8 

22.83 

18 

32.80 

28 

39- 5^ 

38 

44-78 

48 

49.19 

9 

24.11 

19 

33-57 

29 

40 .  10 

39 

45.26 

49 

49.60 

lO 

25-30 

20 

34-31 

30 

40.65 

40 

45-72 

50 

50.00 

to  know  now,  is  the  length  of  the  several  sections  of  the  par- 
tition of  which — in  the  last  case  of  tone  combination,  4  and 
9 — the  first  or  initial  one  moves  up  and  down  nine  times  and 
produces  the  tone  9,  the  second  produces  no  definite  tone  with 


MECHANICS  OF  THE  INNER  EAR 


77 


certainty,  the  third  produces  the  tone  4,  the  fourth  the  tone 
2,  and  the  fifth  the  tone  1. 

The  fluid  quantity  for  the  tone  9  is  measured,  as  we  found 
above,  by  the  relative  number  nine.  Now,  let  us,  for  exam- 
ple, assume  that  this  means  an  equal  number  of  fluid 
units  in  our  table.  We  then  read  off  the  corresponding  par- 
tition length  as  being  24.11  units.  The  fluid  quantity  for  the 
uncertain  tone  was  measured  as  two  units.  But  now,  we  can- 
not simply  read  off  from  the  table  the  number  of  partition 
units  corresponding  to  two;  for  the  partition  section  making 

TONE  INTENSITIES  IN  THE  COMBINATION  4  AND  9 


Tones 

Uniform  width 

Tapering 

9 

22.5% 

52.7% 

Uncertain 

5.0% 

5.0% 

4 

55 -0% 

34-8% 

2 

2-5% 

X.1% 

I 

15.0% 

6.4% 

room  for  these  two  fluid  units  is  not  an  initial  section.  We 
must  read  off,  therefore,  the  value  corresponding  to  eleven 
fluid  units  (26.42)  and  subtract  from  this  the  value  correspond- 
ing to  nine  fluid  units  (24.11).  We  thus  see  that  the  length 
of  the  partition  section  about  the  tone  of  which  we  could  not 
come  to  a  decision  is  2.31  units.  The  fluid  quantity  for  the 
tone  4  was  measured  as  twenty-two.  But  here  again  we  can- 
not simply  read  off  the  length  of  the  partition  section  pro- 
ducing this  tone,  because  this  section  is  not  an  initial  section. 
We  must  read  off  the  values  for  9+^+22=33  and  for  9+2=11 
and  subtract  the  latter  from  the  former.  These  values  are 
42.28  and  26.42.     The  length  of  that  section  of  the  partition 


78  UNIVERSITY  OF  MISSOURI  STUDIES 

which  moves  up  and  down  four  times  is  therefore  15.86  units. 
The  intensity  of  the  tone  2  is  one  fluid  unit.  The  length  of 
the  partition  section  corresponding  to  this  fluid  unit  is  42.80' — 
42.28c=.52.  The  fluid  quantity  for  the  tone  1  is  six  fluid  units. 
We  have  to  read  off  from  the  table  the  values  corresponding 
to  9+2+221+ 1+6=: 40  and  to  9+2+22+l=:34  fluid  units.  These 
values  are  45.7'2  and  42.80.  The  length  of  that  section  of  the 
partition  which  produces  the  tone  1  is  therefore  2.9'2  units  of 
the  partition. 

The  relative  intensities  of  the  four  tones  9,  4,  2,  and  1, 
would  then  be,  not  as  nine  to  twenty-two  to  one  to  six,  but  as 
24.1  to  15.9  to  .5  to  2.9 ;  and  the  tone  about 
The  relative  Avhich  we  could  not  reach  a  definite  con- 

intensities  of  the  elusion  would  have  the  relative  intensity 
tones  9,  4,  2,  2.3  instead  of  two.     For  the  sake  of  better 

and   1  comparison  let  us  express  the  relative  in- 

tensities in  percentages.  The  table  shows 
in  one  column  the  tone  intensities  in  case  we  regard  the  par- 
tition as  of  uniform  width  and  in  another  column  the  intensi- 
ties in  case  we  regard  the  partition  as  tapering  and  possess- 
ing those  properties  upon  which  the  present  computation  is 
based. 

We  must  not,  of  course,  regard  the  result  found  in  the 
second  column  of  intensities  as  any  more  final  than  that  in 
the  first  column.  We  have  assumed  that 
This  result  ^^^  initial  section  of  the    partition    tapers 

not  final  uniformly  so  that,  the  initial  width  being 

w,  its  width  is  6w  at  a  distance  of  5'Ow.  But 
we  do  not  know  that  it  tapers  just  this  way.  We  have  further 
assumed  that  the  areas  described  by  cross-sections  of  the  par- 
tition in  moving  from  one  limit  of  position  to  the  other,  are 
geometrically  similar.  But  we  do  not  know  whether  they  are 
or  not.  We  have  further  assumed  that  the  total  movement  of 
the  partition  in  this  case  extends  just  to  the  distance  of  45.7'2w. 


MECHANICS  OF  THE  INNER  EAR 


79 


But  this  is  an  arbitrary  assumption,  and  the  results  of  the  ta- 
ble, as  is  shown  farther  below,  would  look  different  if  the  total 
movement  did  not  extend  just  so  far,  but  farther  or  less  far. 
We  must  not,  then,  regard  this  result  as  final,  but  simply  ob- 
serve if  it  tends  to  change  the  relative  intensities  in  such  a 
direction  as  might  correct  the  intensities  which  seemed  some- 
what objectionable.  Now,  we  objected,  first,  to  the  fact  that 
the  higher  of  the  primary  tones  had  such  a  slight  intensity  com- 
pared with  the  lower  one,  22.5  per  cent  compared  with  55.0 
per  cent.  Now  we  see  that  taking  into  account  the  tapering 
of  the  partition  raises  the  intensity  of  the  tone  9  to  52.7  per 
cent  and  lowers  that  of  the  tone  4  to  34.8  per  cent.  As  stated 
before,  these  particular  figures  must  not  be  regarded  as  a  final 
result.  It  is  irrelevant  that  now  the  lower  tone  is  weaker  than 
the  higher.  What  is  important  is  the  fact  that  the  influence 
in  question  is  in  the  direction  in  which  it  must  be  in  order  to 
correct  the  objectionable  features  of  the  former  computation. 
A  further  result  of  this  influence  is  the  reduction  of  the  former 
intensity  of  the  difference  tone  1,  which  we  regarded  as 
rather  high,  from  15.0  per  cent  to  6.4  per  cent — again  a  change 
in  the  desired  direction. 

We  can  obtain  here  a  more  special  insight  in  addition 
to  the  general  insight  into  the  fact  that  tapering  of  the  par- 
tition tends  to  increase  the  intensities  of  the 

„,         ,     .       .  tones  produced  by  the  initial     sections,  to 

The  relative  in-        ,  ^      ,      .  .  .        .    . 

tensities  not  inde-  decrease  the  mtensities  of  the  tones  pro- 
pendent  of  the  duced  by  more  distant  sections  of  the  par- 
absolute  intensity  tition.  More  especially,  we  shall  observe 
of  the  compound  that  the  amount  of  this  increasing  or  de- 
•  sound  creasing  influence  varies  according  as  the 
total  length  of  the  partition  section  set  in 
motion  varies,  that  is,  as  the  total  intensity  of  the  compound 
sound  heard  varies.  Imagine,  for  example,  three  tones,  which 
we  call  A,  B,  and  C,  being  produced  by  successive  sections  of 


8o 


UNIVERSITY  OF   MISSOURI  STUDIES 


the  partition.  Imagine  further  that  the  quantity  of  displaced 
fluid  for  the  tone  A  is  20  per  cent  of  the  total  amount  of  fluid 
displaced  by  the  compound  sound  wave,  that  the  quantity 
for  B  is  50  per  cent,  and  the  quantity  for  C  30  per  cent.  This 
is  a  percentage  which  might  easily  be  found  in  an  actual  case. 
The  pitch  of  the  tones  A,  B,  and  C  is  irrelevant.  The  table 
below  contains  all  the  values  wihich  are  of  interest  to  us,  for 
two  cases.  In  the  first  case  the  actual  fluid  quantities  are 
two,  five,  and  three,  by  assumption ;  in  the  second  case  they 
are  ten,  twenty-five,  and  fifteen.  That  is,  the  stirrup  move- 
ment in  the  second  case  is  of  the  same  form,  but  exactly  five 
times  as  large  as  in  the  first. 


Quantities    of    dis- 
placed fluid 

Length    of  sections 
(absolute  values) 

Length  of   sections 
(percentages) 

A 

2 
lO 

B 

5 

25 

C 

3 
15 

A 
II-3 
25-3 

B 

ID.  2 

i8.o 

C 

3-8 
6.7 

y 

25-3 
50.0 

A 

44-7% 
50.6% 

B 

40.3% 
36.0% 

C 

15.0% 
13-4% 

The  table  shows  that  the  tone  intensities  do  not  increase 
proportionally  to  the  increase  in  the  amplitude  of  stirrup  move- 
ment. The  amplitude  in  the  second  case  is  five  times  that  of 
the  first  case ;  but  the  total  intensity  (S)  of  the  audible  sound 
in  the  second  case  is  less  than  twice  that  of  the  first  case  (50.0 
compared  with  25.3).  The  table  shows  further  that  the  inten- 
sity of  the  tone  A  is  in  the  first  case  44.7  per  cent,  in  the  second 
case  50.6  per  cent.  That  is  the  increase  in  the  intensity  of  the 
whole  sound  is  favorable  to  the  relative  intensity  of  the  tone 
produced  by  the  initial  section  of  the  partition.  The  percentage 
intensity  of  this  tone.  A,  is  increased  at  the  cost  of  the  tones 
B  and  C,  the  percentages  of  both  of  which  are  diminished. 


MECHANICS  OF  THE  INNER  EAR  8 1 

Thus  far  we  have  studied  the  effect  upon  the  relative  tone 
intensities  of  initial  and  more  distant  sections  which  would 
result  from  a  uniform  increase  in  width  of 
Increase  in  width    ^^^  partition  as  compared  with  a  uniform 
of  partition  width.     But  we  know  that  the     partition 

not  uniform  does     not     increase    uniformly,     but    rapid- 

ly at  first,  near  the  windows,  and  more 
slowly  the  farther  we  go  from  the  windows  (Fig.  24).  To 
understand  the  theoretical  result  of  this  manner  of  increase, 
it  is  not  necessary  to  compute  a  new  table.  It  is  plain  that, 
if  a  more  distant  section  increases  less  than  we  assumed  in 
computing  the  preceding  table,  showing  the  corresponding 
values  of  m  and  x,  this  would  cause  a  longer  piece  of  this  dis^- 
tant  part  of  the  partition  to  move  in  order  to  make  room  for 
a  certain  quantity  of  displaced  fluid.  That  is,  the  decrease  in 
the  broadening  of  the  partition  would  counteract  the  effect 
last  discussed.  We  saw  in  the  preceding  paragraph  that  an 
increase  in  the  intensity  of  the  whole  sound  does  not  leave 
the  relative  intensities  of  the  partial  tones  unaltered,  but  favors 
the  intensities  of  the  tones  on  the  initial  sections,  reduces 
those  on  the  distant  sections.  But  now,  if  we  increase  the  in- 
tensity of  the  whole  sound,  we  throw  the  tones  of  the  more 
distant  sections  on  still  more  distant  sections,  that  is,  on  sec- 
tions where  the  broadening  of  the  partition  is  much  less  than 
that  assumed  in  the  table.  Consequently,  the  tones  of  distant 
sections  cannot  lose  in  percentage  as  much  as  a  derivation 
from  the  table  would  indicate,  but  might  even  gain  somewhat 
in  percentage  of  intensity  through  an  increase  of  the  total  in- 
tensity of  the  sound. 


82  UNIVERSITY  OF  MISSOURI  STUDIES 

The  preceding  paragraphs  must  impress  us  with  the  per- 
plexity of  our  situation.     We  want  to  comprehend  the  facts 
of  audition  as  depending  on  the  structure 

^,  ,     r  and  function  of  the  sense  organ.    But  every 

The  need  of  a  .  . 

more  accurate  endeavor  to  enter  into  the  details  of  the 

and  detailed  function  of  the  organ  is  thwarted  by  the 

anatomical  poverty  and  inaccuracy  of  our  anatomical 

knowledge  knowledge.     We  cannot  obtain  a     definite 

idea  of  the  intensities  of  the  various  physi- 
ological processes  resulting  from  a  compound  aerial  wave  un- 
less we  know  exactly  the  manner  of  increase  in  width  of  the 
partition.  It  is  not  sufficient  to  know  that  it  increases  first 
rapidly,  then  slowly.  We  need  a  very  exact  measurement 
of  the  width  of  succeeding  cross-sections  of  the  partition  and 
of  the  distance  of  each  of  them  from  the  beginning  of  the  par- 
tition near  the  windows. 

On  the  other  hand,  we  need  also  a  much  more  detailed 
and   accurate   comparison   of  the   relative   intensities   of   the 
components  of  stronger  and  weaker  com- 
,p,  J    £  pound  sounds,  based  on  psychological  ex- 

more  accurate  perimentation  and  observation.    Thus  far, 

observation  of  the    practically  nothing  in  this  regard  is  known 
psychological  with  exactness.    It  is  to  be  hoped  that,  in 

facts  of  hearing  spite  of  the  extraordinary  technical  diffi- 
culties and  the  costliness  of  the  apparatus 
required  for  such  investigations,  an  accurate  knowledge  of 
these  psychological  facts  will  be  obtained.  We  need  this 
knowledge  because  some  of  the  constants  contained  in  the 
mechanical  theory  may  never  become  directly  measurable,  for 
example,  the  elastic  properties  of  the  partition,  and,  therefore, 
will  have  to  be  inferred  from  their  psychological  conse- 
quences. 


MECHANICS  OF  THE  INNER  EAR  83 

Two  consequences  of  the  particular  shape  of  the  partition 

which  we  have  just  discussed  in  as  much  detail  as  anatomical 

knowledge  permits  should  be  emphasized. 

Two  important  ^^^  ^'^^  ^^  ^^^'^  '^  °^  ^^^  ^rez^^st  biolog- 
consequences  of  ^^^^  significance.  It  is  certainly  important 
the  partition's  for  the  animal  to     be    very     sensitive    to 

shape.  sound,  that  is,  to  be  able  to  hear  sounds 

Sensitiveness  which   are   very  weak   and   cause   only   a 

ot  the  ear  minute  movement  of  the  stirrup.       Now, 

the  initial  part  of  the  partition  being  ex- 
ceedingly narrow,  even  the  minutest  quantity  of  fluid  dis- 
placed by  the  stirrup  must  spread  considerably  lengthwise 
over  the  partition  and  thus  stimulate  quite  a  number  of  nerve 
ends.  But  it  would  not  be  advantageous  to  have  the  partition 
equally  narrow  all  along.  In  that  case  comparatively  weak 
objective  sounds  would  cause  the  whole  partition  to  move  up 
and  down  and  the  displaced  fluid  for  which  no  room  can  be 
made  by  the  partition,  to  flow  back  and  forth  through  the 
"safety  valve."  Strong  objective  sounds  would  then  make 
the  same  impression  upon  the  animal  as  sounds  of  medium 
physical  intensity.  This  disadvantage  is  overcome  by  the 
partition's  tapering,  by  its  being  narrow  at  the  beginning, 
but  wide  farther  on,  so  that  even  sounds  of  considerable 
strength  do  not  involve  the  whole  partition.  But  again,  there 
would  be  a  disadvantage  if  the  partition's  width  increased  uni- 
formly :  for  then  the  relative  intensities  of  simultaneous  tones 
— as  we  have  seen — would  not  be  even  approximately  inde- 
pendent of  the  absolute  intensity  of  the  total  sound.  This 
disadvantage  might  be  avoided  by  the  width  increasing  first 
rapidly,  then  more  and  more  slowly.  If  it  is  thus  avoided, 
either  partially  or  totally,  we  do  not  exactly  know  because  of  lack 
of  exact  anatomical  data. 


84 


UNIVERSITY  OF  MISSOURI  STUDIES 


The  second  of  the  consequences  to  be  emphasized  is 
probably  of  little  biological  significance,  but  possibly  of  some 
importance  to  the  student  observing  differ- 
Conditions  more  ^"^^  ^°^^^  '^^  ^  psychological  laboratory. 
or  less  favorable  ^^  ^^  quite  possible  that,  as  a  result  of  the 
to  the  observation  tapering  not  being  uniform  but  decreasing 
of  difference  as  the  windows  are  left  behind,  the  rela- 

*^^^^  tive  intensity  of  difference  tones,  which  are 

obviously  produced  by  the  more  distant 
sections  of  the  partition,  is  somewhat  greater  when  the  abso- 
lute intensity  of  the  whole  sound  is  rather  great.  If  this  is 
so,  it  would  be  advisable  to  use  for  the  observation  of  dif- 
ference tones  fairly  strong  primary  tones  rather  than  weak 
ones.  Whether  this  conclusion  is  borne  out  by  experience,  I 
must  leave  to  the  reader  to  decide. 

The  above  discussion  of  tone  intensities  naturally  leads  us 
to  take  up  the  theoretical  aspects  of  the  fact  frequently  ob- 
served by  experimenters  that  in  a  combina- 
The  dis-  ^^°^  ^^  ^  lower  and  a  higher  tone  the  latter 

appearance  of  is  sometimes  entirely  inaudible,  provided, 

a  higher  tone  of  course,  that  it  is  physically  much  weaker 

than  the  former.  The  reverse,  however, 
that  is,  the  disappearance  of  a  physically  weak  low  tone  when 
sounded  together  with  a  strong  higher  tone,  has  hardly  been 
observed.  The  phenomenon  in  question  can,  perhaps,  be  most 
easily  observed  with  such  ratios  at  1:2,  2:3,  or  1:3.  Let  us 
study,  then,  one  of  these  ratios,  say  1  :  2,  from  the  theoretical 
point  of  view. 


Fig.  26.     The  combination  i  and  2,  unequal  amplitudes 


MECHANICS  OF  THE  INNER  EAR  85 

Let  US  combine  two  sinusoids  according  to  the  following 
equation : 

f(x)  =z  2sin;i:+sin2;t:. 
The  combination      ^^^^  ^^'  ^^^  amplitude  of  the  sinusoid  of 
1  and  2,  when  2       the  shorter  period  is  one-half  of  the  am- 
is comparatively      plitude  of  the  sinusoid  of  the  longer  pe- 
"^^^^  riod.    Figure  36  shows  the  curve  represent- 

ing the  stirrup  movement,  and  the  accom- 
panying table  shows  the  exact  numerical  values  of  those  points 
of  the  curve  which,  as  we  shall  see,  are  of  particular  import- 
ance to  us,  that  is,  the  maxima  and  minima,  and  the  points 
of  inflection.  These  values  are  easily  found  in  this  particular 
case.  To  find  the  maxima  and  minima,  we  have  to  set  the 
first  derivative  of  the  above  function  equal  to  zero  and  solve 
the  equation  for  x;  for  the  maxima  and  minima  are  those 
points  where  the  tangential  angle  or  differential  coefficient 
is  zero, 

f(x)  =  2cos;i;  +  2cos2iar  =  0. 

To  find  the  points  of  inflection,  we  have  to  set  the  second 
derivative  equal  to  zero  and  solve  the  equation  for  x;  for  the 
points  of  inflection  are  those  points  of  the  curve  where  the 
tangential  angle  neither  increases  nor  decreases. 
f"(x)  =  —  2sin;r  —  4sin2;r  =  0. 

The  purely  arithmetical  work  I  do  not  care  to  perform 
here.  The  table  shows  its  results.  It  is  plain  that,  if  we  rep- 
resent the  successive  positions  of  the  partition  according  to 
the  same  rules  as  formerly  employed,  we  find  that  only  one 
tone  can  become  audible,  the  tone  1.  The  tone  2  has  disap- 
peared because  its  addition  does  not  increase  the  number  of 
the  maxima  and  minima  of  the  compound  curve  (Fig.  26),  but 
merely  influences  its  shape.  However  interesting  this  in- 
sight may  be  into  the  fact  that  a  weak  higher  tone  added 
to  a  strong  lower  tone  may  be  entirely  inaudible,  the  present 
theoretic  result  is  not  quite  satisfactory.    It  is  somewhat  un- 


86 


UNIVERSITY  OF   MISSOURI  STUDIES 


satisfactory  because  it  seems  improbable  that  the  higher  octave 
should  become  inaudible  as  soon  as  its  amplitude  is  decreased 
to  one-half  of  the  amplitude  of  the  lower  tone.  It  seems,  judg- 
ing from  experimental  experience,  that  the  higher  octave  must 
be  weakened  by  far  more,  in  order  to  become  entirely  inaudi- 


INTERVAL   1:2,  AMPLITUDES   2  :  I 


Ordinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Inf. 

0 

0 

259S 

B 

2598 

Max. 

+  2598 

600 

SI96 

C 

2598 

Inf. 

+  1125 

1045 

3723 

D 

1473 

Inf. 

0 

iSoo 

2598 

E 

1125 

Inf. 

—  1125 

2555 

1473 

F 

II2S 

Min. 

-  2498 

3000 

0 

G  =  A 

1473 

Inf. 

0 

3600 

2598 

B 

1473 

ble.  Now,  to  correct  the  above  theoretic  result,  we  cannot 
make  use  of  the  previous  considerations  concerning  the  in- 
fluence of  the  tapering  of  the  partition.  As  long  as  there  is 
an  initial  section,  however  short,  jerked  down  and  up  twice 
during  the  period,  the  result  of  tapering  may  be  the  length- 
ening of  this  section  and  a  corresponding  increase  of  the  rel- 
ative intensity  of  the  higher  tone.  But  when  there  is  no 
initial  section  at  all  which  moves  twice,  no  tapering  of  the  par- 
tition can  create  one.  Let  us,  therefore,  recall  the  other  pro- 
visional  assumptions. 


MECHANICS  OF  THE  INNER  EAR  87 

The  second  of  our  provisional  assumptions  is  that  the 
partition  is  perfectly  inelastic,  that  is,  not  offering  any  re- 
sistance to  a  displacement  until  either  of 
The  second  ^^^  limits  is  reached,  and  then  offering  ab- 

provisional  solute  resistance.    Now,  does  our  anatom- 

assumption  ical  knowledge  warrant  such    an   assump- 

recalled  tion?    The  most  striking  fact  derived  from 

an  anatomical  study  of  the  organ  is  the 
absence  of  any  solid  body  which  might  serve  to  interfere  sud- 
denly, abruptly,  with  a  yielding  movement  of  the  partition  in 
either  direction.  Even  the  analogy  with  the  leather  seat 
of  a  chair  is  hardly  admissible  if  we  mean  thereby  a  flabby, 
wrinkled  piece  of  leather.  The  analogy  probably  holds  good 
only  if  we  imagine  the  leather  in  such  a  condition  as  we  find 
it  in  a  new,  unused  chair,  occupying  a  perfect  plane,  being 
practically  free,  however,  from  any  stresses  as  long  as  no  weight 
is  resting  upon  it,  yielding  to  a  certain  extent  if  a  certain  weight 
is  placed  upon  it,  but  not  yielding  in  proportion  to  the  weight 
if  the  weight  is  increased.  It  is  probably  in  a  similar  manner 
that  the  partition  resists  pressure.  What  determines  the 
limit  of  yielding  must  be  the  partition's  own  elasticity.  But 
let  us  always  remember  that  there  is  no  elastic  force — no 
stress — in  the  partition  while  in  its  normal  position,  that  its 
elastic  force  is  the  result  of  a  displacement  in  either  direction, 
that  this  elastic  force  increases  much  more  rapidly  than  the 
displacement,  and  that  therefore  a  constant  increase  of  press- 
ure on  any  point  of  the  partition  does  not  cause  a  constant 
movement  of  this  point,  but  a  movement  first  rapid,  then 
quickly  decreasing  in  velocity.  Figure  27  is  a  graphic  repre- 
sentation of  such  a  function  under  the  arbitrary  assumption — 
which,  perhaps,  may  be  regarded  as  a  rough  approximation 
to  the  actual  conditions — that  the  elastic  force  of  the  partition 
increases  proportionally  to  the  tangent  of  its  displacement.  The 
abscissae     represent     the     increasing     pressure,     the     ordinates 


88 


UNIVERSITY  OF  MISSOURI  STUDIES 


the  corresponding  displacements  of  the  partition.  We  notice, 
then,  that  there  is  a  practical  limit  of  yielding,  that  an  increase 
of  pressure  beyond  a  certain  point  is  practically  ineffective,  does 
not  cause  any  further  displacement  to  speak  of. 

There  can  be  no  doubt  that  the  assumption  of  a  relation 
existing  between  the  displacement  of  the  partition  and  the 
pressure,  similar  to  the  relation  between  an  angle  and  its 
tangent — ^however  rough  the  approximation  to  the  facts — is 


■ 

-v3 

c 

E 

/ 

Q) 

O 

f\ 

C3     ■ 

h 

Q_ 

CO 

;i 

pressure 

Fig.  27.     The  probable  relation  between  pressure  and  displacement  of  the  partition 

much  better  adapted  to  the  anatomical  facts  than  the  second 
provisional  assumption.  Of  course,  the  second  provisional  as- 
sumption simplifies  greatly  the  graphic  representation  of  the 
successive  positions  of  the  partition,  but  at  the  cost  of  all 
accuracy.  Wherever  the  approximation  thus  possible  is  suffi- 
cient for  our  purposes,  we  shall,  of  course,  continue  to  work 
under  that  simpler  assumption.  But  let  us  now  apply  the 
latter  assumption  to  our  problem  of  representing  the  succes- 
sive positions  of  the  partition  which  correspond  to  the  stirrup 


MECHANICS  OF  THE  INNER  EAR  89 

movement  of  the  curve  in  figure  26.  Let  us  disregard,  how- 
ever, the  varying  width  of  the  partition,  in  order  to  avoid  too 
much  compHcation.  We  shall  again  assume  the  partition  to  be 
of  uniform  width,  without,  however,  forgetting  the  fact  that 
this  is  an  arbitrary  simplification  of  the  conditions. 

Imagine  that  the  whole  partition  is  in  its  normal  posi- 
tion, free  of  any  stress,  and  that  the  stirrup  begins  an  outward 
movement  of  the  form  of  the  curve  from  E 
The  significance      *°  ^  ^^  figure  26.    We  see  from  the  curve 
of  a  point  of  that  the  stirrup  moves  at  first  very  slowly, 

inflection  then  gradually  more  and  more  quickly  un- 

til at  F,  the  point  of  inflection,  it  moves 
with  the  greatest  velocity.  Now,  a  simple  consideration  will 
make  it  plain  to  us  that  the  pressure  acting  upon  the  initial  part 
of  the  partition  must  be  dependent  on,  probably  be  propor- 
tional to  the  velocity  of  the  stirrup.  If  the  velocity  of  the  stir- 
rup movement  were  extremely  small,  no  point  of  the  partition 
would  move  more  readily  than  any  other,  and  consequently 
none  of  them  would  move  to  a  considerable  extent;  but  the 
fluid  would  every  time  and  all  the  time  flow  through  the 
opening  at  the  end  of  the  tube  which  we  called  the  safety 
valve,  because  there  would  then  be  practically  no  friction  at 
any  point  within  the  tube,  and  an  infinitesimal  elastic  force 
of  displacement  could  keep  the  partition  in  place.  On  the  other 
hand,  if  the  velocity  of  the  stirrup  movement  is  not  very 
small,  the  points  of  the  partition  near  the  windows  receive 
the  greatest  push  from  the  fluid,  farther  points  only  a  slighter 
push,  very  quickly  diminishing  with  increasing  distance,  and 
at  some  distance  away  the  push  could  be  regarded  as  practi- 
cally infinitesimal ;  all  this  as  the  result  of  the  friction  of  the 
fluid  in  the  narrow  tube,  the  total  influence  of  which  is  the  greater 
the  longer  the  column  of  fluid  in  question,  measuring  this  column 
from  the  windows. 


90 


UNIVERSITY  OF  MISSOURI  STUDIES 


As  the  stirrup  moves  away  from  E,  the  initial  part  of  the 
partition  yields  upwards,  as  shown  in  figure  28  at  I.  By  I, 
II,  and  so  forth,  are  meant  successive  moments  between  E 
and  G  in  figure  26.  The  increasing  velocity  of  the  stirrup  re- 
sults at  II  in  an  increased  pressure  at  all  the  points  of  the 
partition  which  had  yielded  at  I.  Therefore,  at  II  in  figure  38 
these  points  are  somewhat  farther  displaced  than  they  were  at 
I,  but  not  proportional  to  the  increase  of  the  velocity  of  the 
stirrup  but  much  less,  according  to  figure  27.     At  the  same 


/ 
// 
/// 

r 

IV 
V 


VI  c 


Fig.  28.     Seven  successive  positions  of  the  partition,  three  preceding 
and  three  following  an  inflection  point  (F) 


time  we  notice  that  the  part  of  the  partition  which  has  now 
yielded  extends  much  farther  to  the  right  at  II  than  at  I; 
for  the  stirrup  has  displaced  much  more  fluid  at  II  than  at  the 
earlier  moment  I,  and  the  slight  increase  in  the  displacement 
of  those  parts  of  the  partition  which  were  already  displaced 
at  I,  can  not  nearly  make  room  for  all  this  fluid.    Therefore  the 


MECHANICS  OF  THE  INNER  EAR  9 1 

Spreading  of  the  displacement  lengthwise  over  the  partition. 
At  III  the  velocity  of  the  stirrup  is  still  greater  than  at  II. 
Therefore  we  notice  again  a  slight  increase  in  the  displace- 
ment of  the  initial  part  of  the  partition.  But  as  the  stirrup 
approaches  F,  this  increase  of  displacement  of  the  initial  parts 
must  become  less ;  for  the  velocity  of  the  stirrup  is  now  nearly 
constant,  its  increase  very  slight,  and  the  increase  of  displace- 
ment is  in  any  case  much  less  than  proportional  to  the  increase 
of  velocity,  according  to  figure  27.  As  soon  as  the  stirrup  passes 
F,  its  velocity  begins  to  decrease.  Immediately  the  press- 
ure on  the  whole  piece  of  the  partition  which  has  yielded  de- 
creases; and  this  whole  piece,  therefore  begins  to  move 
slowly  back  by  its  elasticity  in  the  direction  of  its  normal 
position.  It  is  clear,  however,  from  figure  27  that  even  a 
considerable  decrease  of  the  velocity  of  the  stirrup  causes  only 
a  slight  decrease  of  the  displacement  until  the  stirrup  ap- 
proaches G,  when  its  velocity  approaches  zero  and  the  part 
of  the  partition  in  question  can  move  more  rapidly  by  its 
elasticity  since  it  has  no  longer  to  overcome  much  pressure 
caused  by  the  stirrup.  It  does  not  follow,  however,  that  any 
point  of  the  partition  has  returned  to  its  normal  position  by 
the  time  the  stirrup  reaches  G.  The  initial  sections  have 
merely  moved  in  the  direction  of  their  normal  position.  And 
meanwhile,  new  points  of  the  partition  to  the  right  must 
have  yielded  upwards  to  make  room  for  the  fluid  being  dis- 
placed all  the  time  by  the  stirrup  in  moving  towards  G. 
Three  positions  of  the  partition  between  F  and  G  are  shown 
in  figure  28  at  IV,  V,  and  VI. 


92 


UNIVERSITY  OF  MISSOURI  STUDIES 


One  of  the  consequences  of  the  decrease  of  pressure  on  the 
partition  at  the  point  of  inflection  between  a  maximum  and  a 
preceding  or  following  minimum  of  the 
Theoretic  con-  curve  consists  in  the  fact  that  the  partition 

sequences  of  does  not  move  up  and  down  so  suddenly 

the  inflection  as  it  appeared  from  our  previous  graphic 

of  the  curve  representations.     We  had  to  point  out  this 

fact  before  in  mentioning  the  irregular- 
ity with  which  stimuli  often  seem  to  be  received  by  the  nerve 
ends  according  to  our  simplified  graphic  representation.  The 
exact  time  when  a  stimulus — a  shock,  as  we  called  it — is  re- 
ceived we  now  find  to  be  dependent  also  on  the  location  of  each 
inflection  point,  not  merely  on  the  temporal  location  of  the 
maxima  and  minima.  Unfortunately,  however,  we  can  not 
determine  the  time  of  each  shock  with  certainty  even  now, 
taking  into  account  the  inflection  point.  This  important  ques- 
tion of  theoretical  detail  must  be  left  open  for  future  investi- 
gation. 

Another  consequence  of  the  decrease  of  pressure  on 
the  partition  marked  by  any  point  of  inflection  consists  in  the 
fact  that  a  double  movement — up  and  down — of  the  partition 
may  result,  not  only  frorh  an  alternation  of  maxima  and  mini- 
ma of  a  curve,  but  also  from  an  alternation  of  inflection  points 
marking  an  increasing  and  decreasing  velocity  of  the  stirrup. 
This  means  that  the  number  of  shocks  received  by  the  nerve 
ends  during  one  period  of  the  curve  may  exceed  the  total  num- 
ber of  maxima  (or  minima)  in  case  any  part  of  the  curve 
from  a  maximum  to  a  minimum  or  from  a  minimum  to  a  max- 
imum contains  more  than  a  single  point  of  inflection.  An  example 
will  be  given  at  once. 


MECHANICS  OF  THE  INNER  EAR  93 

Let    us    return    to    the    theoretical  analysis    of    the    whole 

curve  in  figure  26.     From  A  to  C  the  stirrup  moves  inwards, 

pushing  down  a  certain  length  of  the  parti- 

The  successive  ^^°"-  ^^^  '""'^'^^  P^^^  °^  ^^'^  ^^^S^^'  ^^W" 
positions  of  the  ever,  begins  a  slow  upward  movement  as 
partition  corres-  soon  as  the  velocity  of  the  stirrup  begins 
ponding  to  to  decrease,  at  B.     The  same  part  moves 

ngure  »o  ^p  n-^ore  quickly  when,  at  C,  the  stirrup 

reverses  its  movement  and  begins  to  pull 
it  upward.  We  therefore  see  at  B  in  figure  39  the  initial  two 
sections  in  an  extreme  downward  position.     At  C,  we  see  them 

3^ ^ .,. 

g, ^- 

PL ! ^-, 

£t 4 ■-,— 

rL -— -^ .— 

g( L.-- 

2  111 

Fig.  29.     The  combination  i  and  2.     Compare  figure  26 

only  in  a  medium  downward  position,  and  at  the  same  time 
we  find  the  following  two  sections  of  the  partition  in  a  similar 
downward  position  since  the  stirrup  has  continued,  from  B  to 
C,  to  move  inwards.  It  is  plain  that  to  take  into  account,  in 
our  graphic  representation,  only  two  kinds  of  displacements  in 
either  direction,  an  extreme  and  a  medium  one,  is  again  an  ar- 
tificial simplification,  introduced  merely  to  suit  our  momentary 
needs,  in  spite  of  the  fact  that  thus  we  lose  sight  of  some  of  the 
details  of  the  movement.  Actually,  the  movement  probably 
occurs  rather  in  the  form  of  figure  28.  But  the  simplification 
used  in  figure  29  not  only  renders  the  drawing  of  the  figure 


94  UNIVERSITY  OF  MISSOURI  STUDIES 

easier,  but  also  contributes  towards  a  readier  comprehension 
of  the  significance  of  the  graphic  representation,  towards  a 
quicker  reading  off  of  the  tones  to  be  heard. 

At  D  we  see  the  first  section  in  an  extreme  upward  position 
since  the  stirrup  has  moved  outwards  and  has  reached  a  max- 
imum velocity.  At  E,  the  first  section  has  returned  to  a  me- 
dium displacement  since  the  velocity  of  the  stirrup  has  reached 
a  minimum.  At  the  same  time  the  second  section  of  the  par- 
tition has  moved  upwards  as  a  result  of  the  continued  outward 
movement  of  the  stirrup.  At  F  we  find  the  initial  three  sec- 
tions of  the  partition  in  an  extreme  upward  position;  for  the 
stirrup  has  continued  to  move  outwards  and  has  also  reached 
a  maximum  of  velocity.  At  G  all  four  initial  sections  of  the 
partition  are  in  an  upward  position  since  the  stirrup  has  con- 
tinued to  move  outwards.  But  they  are  only  in  a  medium 
displacement  since  the  velocity  of  the  stirrup  has  again  reach- 
ed a  minimum. 

Looking  now  over  the  four  columns  in  figure  29,  we  notice 
that  the  first  shows  an  extreme  upward  position  of  this  section 

of  the  partition  at  F,  a  medium  upward 
De  we  hear  position    at    G=rA,    an    extreme    downward 

both  tones  position  at  B,  a  medium  downward  posi- 

2  and  1?  tion  at  C,  an  extreme  upward  position  at 

D,  a  medium  upward  position  at  E,  an  ex- 
treme upward  position  again  at  F.  This  section  of  the  parti- 
tion, therefore,  has  moved  up  and  down  twice  during  the  pe- 
riod, the  second  upward  movement  occurring  between  E  and 
F.  It  is  quite  probable,  then,  that  the  nerve  ends  located  on 
this  section  receive  two  shocks  during  the  period.  The  second 
section  of  the  partition  has  an  extreme  upward  position  at  F, 
a  medium  upward  position  at  G=A,  an  extreme  downward 
position  at  B,  a  medium  downward  position  at  C  and  D, 
and  a  medium  upward  position  at  E.  It  follows  that 
this  section  moves  up  and  down  only  once  during  the  pe- 


MECHANICS  OF  THE  INNER  EAR 


95 


riod,  and  that  the  nerve  ends  located  there  receive  only  one 
shock  during  the  period.  The  third  section  has  an  extreme 
upward  position  at  F,  a  medium  upward  position  at  G=:A  and 
also  at  B,  a  medium  downward  position  at  C,  D,  and  E.  The 
nerve  ends  of  this  section  receive  therefore  one  shock  during 
the  period.  The  fourth  section  has  a  medium  upward  position 
at  G=A  and  at  B,  a  medium  downward  position  at  C,  D,  E,  and 
F.  The  nerve  ends  of  this  section  receive  therefore  one  shock 
during  the  period.  It  is  plain,  then,  that  from  our  theory  we 
must  expect  to  hear  the  tone  2  as  well  as  the  tone  1,  the  former 
conveyed  by  the  first,  the  latter  by  the  three  following  sec- 
tions of  the  partition. 

To  determine  the  relative  intensities  of  the  tones  heard, 
we  have  to  compare  the  length  of  the  initial  section  of  the  par- 
tition with  the  total  length  of  the  three 
Sixth  provision-     following   sections   when   added   together. 
al  assumption  For  simplicity's  sake,  let     us    make    this 

comparison  again  under  the  third  and 
fourth  provisional  assumptions,  and  also  under  a  new  assump- 
tion, namely,  that  the  fluid  for  which  room  is  made  or  whose 
room  is  taken  by  a  move  of  the  partition  from  a  medium  to  an 
extreme  (or  the  reverse)  displacement  on  the  sam^  side  (eith- 
er above  or  below  the  normal  position)  is  a  negligible  quanti- 
ty. That  this  assumption  simplifies  our  representation  of  the 
successive  positions  of  the  several  sections  of  the  partition  is 
clear,  since  we  may  thus  take  the  length  of  each  section  pro- 
portional to  the  ordinate  difference  of  the  corresponding  points 
of  the  curve.  For  instance,  the  third  and  fourth  sections  in 
figure  29,  which  move  down  at  C,  would  be  longer  than  pro- 
portional to  the  ordinate  difference  of  the  points  B  and  C  in 
figure  26  if  the  fluid  displaced  by  the  first  and  second  sections 
in  moving  from  an  extreme  position  at  B  to  a  medium  displace- 
ment at  C  were  not  a  negligible  quantity.  In  the  latter  case, 
the  fluid  displaced  by  the  first  and  second  sections  during  the 


96  UNIVERSITY  OF   MISSOURI  STUDIES 

time  from  B  to  C  would  have  to  be  made  room  for  by  the  third 
and  fourth  sections,  which,  then,  by  necessity  would  extend 
farther  to  the  right  than  in  proportion  to  the  stirrup  movement 
from  B  to  C.  To  take  this  into  account  would  extraordinarily 
complicate  the  graphic  representation  without  offering,  at 
present,  a  correspondingly  great  advantage.  This  additional 
extension  of  the  third  and  fourth  sections  to  the  right  could 
be  but  slight  since  the  amount  of  fluid  in  question  would  be 
but  slight.  This  becomes  clear  from  a  glance  at  figure  27. 
We  have  learnt  from  this  figure  that  some  pressure  added  to 
a  given  pressure  does  not  cause  a  proportional,  but  a  much 
smaller  increment  to  be  added  to  the  previous  displacement  of 
the  partition;  and  thus  the  amount  of  fluid  in  question  may  be 
entirely  neglected  without  depriving  us  of  the  right  to  regard 
our  representation  as  an  approximation  to  the  actual  positions 
of  the  partition   sections. 

We  may,  then,  under  the  third,  fourth,  and   sixth  pro- 
visional assumptions,  regard  the   relative    intensities    of    the 
tones  as  proportional  to  the   ordinate    dif- 
The  relative  ferences  in  the  table  belonging  to  figure 

intensities  of  2^-     We  find  in  the  table  the  value  1473  as 

2  and  1  expressing  the  ordinate  difference  of  C  and 

D,  the  value  1125  of  D  and  E,  1125 
of  E  and  F,  and  1473  of  F  and  G,  the  sum  of  these  last  three  being 
372*3.  Therefore,  under  the  above  simplifying  assumptions, 
the  relative  intensity  of  the  tone  2  compared  with  1  is  about 
as  fifteen  to  thirty-seven. 

Let  us  now  apply  our  theory  to  the  ratio  of  the  vibration 
rates  5 :  8.     The  curve  in  figure  30  represents  the  function 
f(x)  ■=  sin5.r  -\-  sinS.*", 

The  table  below    contains    all    the    abscissa    and    ordinate 

_,  ,  .  values  of  the  maxima  and  minima  as  well 

The  combma-  ,       .   ,,      . 

tion  5  and  8  ^^    °^  inflection    points    of    the    curve. 

Equal  ampli-  The   inflection   points   are   computed  as   the 

tudes  of  stirrup  maxima  and  minima  of  the  first  derivative 

movement  curve,  represented  by  the  function 


MECHANICS  OF  THE  INNER  EAR 


97 


f'(x)  =  5cos5i;«r  +  8cos8;ir. 
It  is   impossible,   in  this   case,  to  apply  the  simple  method   of 


Fig.  34 


Fig.  32       Bi 


Fig.  30       S, 


Fig.  36 


Fig.  38 


The  combination  5  and  8  with  different  amplitude  ratios 


9» 


UNIVERSITY  OF  MISSOURI  STUDIES 


finding  the  corresponding  ordinate  and  abscissa  values  of  the 
maxima  and  minima  of  these  two  functions  by  making  their 
derivatives  equal  to  zero  and  solving  the  resultant  equations 

Interval  5:8,  Equal  Amplitudes 


Inf. 

Max. 

Inf. 

Min. 

Inf. 

Max. 

Inf. 

Min. 

Inf. 

Max. 

Inf. 

Min. 

Inf. 

Max. 

Inf. 

Min. 

Inf. 

Max. 

Inf. 

Min. 

Inf. 

Max. 

Inf. 

Min. 

Inf. 

Max. 

Inf 

Min. 

Inf. 

Max. 

Inf. 

Min. 

Inf. 


Ordinate 


-f  188 

+  24 

—  100 

—  .51 
+  3 

—  29 

—  61 
+  59 
+  167 

—  18 

—  199 

—  36 
+  137 
+  61 

—  26 

o 

-f  26 

—  61 

—  137 
+  36 
+  199 
+  18 

—  167 

—  59 
+  61 
+  29 

—  3 
+  51 
-j-  100 

—  24 


Abscissa 


o 

131 
249 

385 
474 
576 
661 
740 
872 

983 
1116 
1244 
1367 
1504 
1603 

1725 
1800 

187.5 
1997 
2096 
2233 
2356 
2484 
2617 
2728 
2860 

2939 
3024 
3126 
3215 
3351 
3469 
3600 


Ordinate 

199 

V 

387 

W 

223 

X 

99 

Y 

148 

Z 

202 

SI 

170 

83 

138 

e 

258 

S) 

366 

e 

181 

& 

0 

A 

163 

B 

336 

C 

260 

D 

173 

E 

199 

F 

225 

G 

138 

H 

62 

I 

235 

J 

398 

K 

217 

L 

32 

M 

140 

N 

260 

0 

228 

P 

196 

Q 

250 

R 

299 

S 

175 

T 

II 

U 

199 

V 

Ordinate 
Difference 


188 

1 88 

164 

124 

49 

54 

32 

32 

120 

108 

185 
181 

163 

173 
76 

87 
26 
26 

87 
76 

173 

163 

181 

185 

108 

120 

32 

32 

54 

49 

124 

164 


MECHANICS  OF  THE  INNER  EAR 


99 


for  X.  This  is  impossible  because  the  equations  to  be  solved 
would  be  of  the  eighth  degree.  We  have  to  use,  therefore, 
the  only  method  left,  however  great  our  sacrifice  of  time,  and 
to  calculate  directly  a  sufficiently  large  number  of  values  from 
which  we  then  select  the  largest  and  smallest.  In  this  way  the 
values  of  the  table  have  been  obtained.  By  adding  199'  to  each 
of  the  values  of  the  first  column  we  get  the  third  column, 
which  offers  the  advantage  of  containing  only  positive  ordi- 
nates.  This  procedure  is  equivalent  to  selecting  a  different 
horizontal  coordinate,  which  is  always  dependent  on  our 
choice.  The  ordinate  value  zero,  thus  obtained,  is  the  one  which 
belongs  to  point  A  in  figure  30.  The  successive  positions  of 
the  partition  corresponding,  under  the  sixth  provisional  as- 
sumption, to  all  the  maxima,  minima,  and  inflection  points  of 
the  curve  are  shown  in  figure  31. 

Let  us  at  once  examine  the  movements  of  the  three  sec- 
tions, the  fiftieth,  the  fifty-first,  and  the  fifty-second.*  At  A,  we 
find  these  sections  occupying  a  medium  up- 
What  tones  do  ward  position.  From  A  to  B  they  move  down. 
we  hear?  The  From  B  to  C  they  begin  to  move  up. 
tone  8  From  C  to  D  they  continue  to  move  up. 

From  D  to  E  they  begin  to  move  down 
and  continue  to  move  down  until  G.  From  G  to  H  they  move 
up,  completing  thus  the  second  down  and  up  movement. 
From  H  to  J  they  move  down,  and  from  J  to  L  up,  complet- 
ing the  third  down  and  up  movement.  From  L  to  N  they  move 
down,  and  from  N  to  Q  up,  completing  the  fourth  down  and 
up  movement.  From  Q  to  R  they  move  down,  and  from  R  to  T 
up,  completing  the  fifth  down  and  up  movement.  From  T  to 
V  they  move  down,  and  from  V  to  X  up,  completing  the  sixth 


♦  For  a  perfect  understanding  of  the  details,  the  reader  will  have  to 
draw  figure  31  (and  the  similar  figures  following)  for  himself  on  a  larger 
scale,  and  to  inscribe  the  exact  values  as  derived  from  each  corresponding 
table. 


lOO 


UNIVERSITY  OF  MISSOURI  STUDIES 


MECHANICS  OF  THE  INNER  EAR  lOi 

dowm  and  up  movement.  From  X  to  21  they  move  down, 
and  from  9t  to  ©  up,  completing  the  seventh  down  and  up 
movement.  From  ©  to  ®  they  move  down,  and  from  S  to  g 
up,  completing  the  eighth  down  and  up  movement.  From  g 
to  Qf)=A  they  begin  to  move  down  and  continue  to  move 
down  after  A,  as  we  have  seen. 

The  movements  of  the  forty-nine  initial  sections  are  so  sim- 
ilar to  those  of  the  three  sections  just  discussed  that  we  convince 
ourselves  easily  that  the  nerve  ends  located  there  receive  the 
same  number  of  shocks  during  the  period. 

The  fifty-third  and  fifty-fourth  sections  move  down  from  g 
to  B,  and  up  from  B  to  D.  Dowm  from  D  to  G,  and  up  from  G 
to  H.  Down  from  H  to  J,  and  up  from  J  to  L.  Down  from 
L  to  N,  and  up  from  N  to  Q.  Down  from  Q  to  R,  and  up  from 
R  to  T.  Down  from  T  to  V,  and  up  from  V  to  X.  Down 
from  X  to  21,  and  up  from  2t  to  ©.  Dbwn  from  ©  to  ®,  and 
up  from  2)  to  g.  The  nerve  ends  located  on  these  sections 
therefore  receive  eight  shocks  during  the  period. 

The  ten  sections  from  the  fifty-fifth  to  the  sixty-fourth  move 
down  from  ^^  to  B,  and  up  from  B  to  D.  Down  from  D  to  G, 
and  up  from  G  to  H.  Down  from  H  to  J,  and  up  from  J  to  L. 
Down  from  L  to  N,  and  up  from  N'  to  Q.  Down  from  Q  to  S, 
and  up  from  S  to  T.  Down  from  T  to  V,  and  up  from  V  to 
X.  Dow^n  from  X  to  21,  and  up  from  21  to  ©.  Down  from  © 
to  2),  and  up  from  2)  to  i^.  The  nerve  ends  located  on  these 
sections  therefore  receive  eight  shocks  during  the  period. 

The  twelve  sections  from,  the  sixty-fifth  to  the  seventy-sixth 
move  down  from  g  to  B,  and  up  from  B  to  D.    Down  from  D 
to  G,  and  up  from  G  to  H.    Down  from  H 
The  tone  6  to  J,  and  up  from  J  to  L.     Down  from  L 

to  N^  and  up  from  N  to  T.     Down  from 
T  to  V,  and  up  from  V  to  X.    Down  from  X  to  ®,  and  up  from 
2)  to  i^.    The  nerve  ends  located  on  these  sections  therefore  re- 
ceive six  shocks  during  the  period. 
The  twenty  seven  sections  from  the  seventy-seventh  to  the 


I02  UNIVERSITY  OF   MISSOURI  STUDIES 

hundred  and  third  move  down  from  g  to  C,  and  up  from  C  to 
H.  Down  from  H  to  J,  and  up  from  J  to 
The  tone  5  L.     Down  from  L  to  N,  and  up  from  N  to 

T.  Down  from  T  to  V,  and  up  from  V  to 
X.  Down  from  X  to  2),  and  up  from  2)  to  g.  The  nerve  ends 
located  on  these  sections  therefore  receive  five  shocks  during 
the  period. 

The  five  sections  from  the  hundred  and  fourth  to  the  hun- 
dred and  eighth  move  down  from  ^^  to  C,  and  up  from  C  to  H. 
Down  from  H  to  J,  and  up  from  J  to  L.  Down  from  L  to  N, 
and  up  from  N  to  T.  Down  from  T  to  V,  and  up  from  V  to 
X.  Down  from  X  to  S),  and  up  from  3)  to  g.  The  nerve  ends 
located  on  these  sections  therefore  receive  five  shocks  during 
the  period. 

All  the  following  sections  to  the  two  hundred  and  sixty- 
seventh  move  down  and  up  five  times  during  the  period.  Let 
us  study  in  detail  only  the  movements  of  the  last  few  of  this 
group.  The  sections  from  the  two  hundred  and  twenty-eighth  to 
the  two  hundred  and  sixty-seventh  move  down  from  A  to  C, 
and  up  from  C  to  I.  Down  from  I  to  K,  and  up  from  K  to 
M.  Down  from  M  to  S,  and  up  from  S  to  U.  Down  from  U 
to  W,  and  up  from  W  to  Y.  Down  from  Y  to  @,  and  up  from 
a  to  @r=A.  The  nerve  ends  located  on  these  sections  there- 
fore receive  five  shocks  during  the  period. 

The  seven  sections  from  the  two  hundred  and  sixty-eighth 
to  the  two  hundred  and  seventy-fourth  move  down  from  Y 
to  C,  and  up  from  C  to  I.  Down  from  I 
The  tone  3  to  K,  and  up  from  K  to  M.     Down  from 

M  to  W,  and  up  from  W<  to  Y.  The  nerve- 
ends  located  on  these  sections  therefore  receive  three  shocks 
during  the  period. 

The  fourteen  sections  from  the  two  hundred  and  seventy- 
fifth  to  the  two  hundred  and  eighty-eighth  move  down 
from  Y  to  C,  and  up  from  C  to  M.  Down 
The  tone  2  from  M  to  W,  and  up  from  W  to  Y.     The 

sections  from  the  two  hundred  and  eighty- 
ninth  to  the  three  hundred  and  thirty-sixth  move  down  from 


MECHANICS  OP  THE  INNER  EAK 


103 


A  to  C,  and  up  from  C  to  M.  Down  from  M  to  W, 
and  up  from  W  to  ®=A.  The  sections  from'  the 
three  hundred  and  thirty-seventh  to  the  three  hundred 
and  sixty-sixth  move  down  from  A  to  K,  and  up  from  K  to  M. 
Down  from  M  to  W,  and  up  from  W  to  @=A.  The  sections 
from'  the  three  hundred  and  sixty-seventh  to  the  three  hun- 
dred and  seventy-sixth  move  down  from  A  to  K,  and  up  from 
K  to  U.  Down  from  U  to  W,  and  up  from  W  to  ®=:A.  All 
these  sections  therefore  receive  two  shocks  during  the  period. 
The  sections  from  the  three  hundred  and  seventy-seventh 
to  the  three  hundred   and  eighty-seventh   move  down   from  U 

to  K,  and  up  from  K  to  U.  The  sections 
The  tone  1  from  the  three  hundred  and  eighty-eighth 

to    the    three    hundred    and    ninety-eighth 

move  down  from  A  to  K,  and  up  from  K    to    @r=:A.     All 

these  sections  therefore  receive  one  shock  during  the  period. 

The  relative  intensities  of  the  several  tones,  if  we  accept 

the  third,  fourth,  and  sixth  provisional  assumptions  for  this 

case,  are  shown  in  the  following  table. 
The  relative  which    contains   the   number    of   partition 

intensities  sections  conveying  each  tone  in  absolute 

numbers  as  well  as  in  percentages. 


Tones 

8 

6 

5 

3 

2 

I 

Intensities 

64 

12 

191 

7 

102 

22 

Percent- 

16.1 

30 

48.0 

1.8 

25.6 

5-5 

ages 

Let  us  now  apply  our  theory  to  the  same  ratio  of  the 
vibration  rates,  but  with  different  amplitudes  of  the  two  sin- 
usoids.   The  curve  in  figure  32  represents 
the  function 

f(x)  =  2sin5Ar  -|-  sin8^. 
This  signifies  that  the  stirrup  movement 
eight  has  an  amplitude  which  is  only  one- 
half  of  the  amplitude  of  the  stirrup  move- 
The  table  below  contains   all   the    abscissa    and 


The  combination 
5  and  8.     The 
amplitude  of 
8  is  decreased 


ment  five. 


I04 


UNIVERSITY  OF  MISSOURI  STUDIES 


ordinate  values  of  the  maxima  and  minima  and  of  the  inflec- 
tion points  of  the  curve. 

Interval  5:8,  Amplitudes  2:1 


Prdinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Inf. 

0 

0 

298 

V 

281 

Max. 

+  281 

142 

579 

W 

281 

Inf. 

+  87 

268 

385 

X 

194 

Min. 

—  143 

436 

155 

Y 

330 

Inf. 

—  118 

512 

180 

Z 

25 

Max. 

St 

Inf. 

—  82 

636 

216 

58 

36 

Min. 

e 

Inf. 

+  no 

846 

408 

® 

192 

Max. 

+  248 

962 

546 

@ 

138 

Inf. 

—  34 

nil 

264 

g 

282 

Min. 

—  298 

1247 

0 

A 

264 

Inf. 

-  62 

1379 

236 

B 

236 

Max. 

+  200 

1535 

498 

C 

262 

Inf. 

+  120 

1638 

418 

D 

80 

Min. 

E 

Inf. 

0 

1800 

298 

F 

120 

Max. 

G 

Inf. 

—  120 

1962 

178 

H 

120 

Min. 

—  200 

2065 

98 

I 

80 

Inf. 

+  62 

2221 

360 

J 

262 

Max. 

+  298 

2353 

596 

K 

236 

Inf. 

-f  34 

2489 

332 

L 

264 

Min. 

—  248 

2638 

50 

M 

282 

Inf. 

—  no 

2754 

188 

N 

138 

Max. 

0 

Inf. 

+  82 

2964 

380 

P 

192 

Min 

Q 

Inf. 

+  118 

3088 

416 

R 

36 

Max. 

+  143 

3164 

441 

S 

25 

Inf. 

-  87 

3332 

211 

T 

230 

Min. 

—  281 

3458 

17 

U 

194 

Inf. 

0 

3600 

298 

V 

281 

MECHANICS  OF  THE  INNER  EAR  IO5 

These  values  have  been  computed  in  the  same  manner 
as  in  the  case  immediately  preceding.  The  successive  po- 
sitions of  the  partition  corresponding,  under  the  sixth  pro- 
visional assumption,  to  the  maxima,  minima,  and  inflection  points 
of  the  curve  are  shown  in  figure  33. 

O  O  g  o  O  o  o 

O  O  O  O  O  O 

M  N  «*5  •><-  lO  >S 

C, - ^ fr^^rrr^   ■ 

DUIZZZIV ^ ^ ... 

/"l 1 1 X.- 

H\. -I— ^ ■^z^rr.:^^, 

/l ■! 4 1., 

Jr y 1 + -■- 

K. ^, 

/.L 4,- ,„ 

y\/L 1.--..- 

N,  y H::^- 

^r f— -^— - 

S. 1 

7L U— -A 

t/, ^ H^- 

W. — ' ^1— 

XL 4 ^ ^.- 

Y\ 1 -P^-,— 

Z^ 1 -FLr— 

fJe:---). 1 Fbj— 

S, -I -h -Fbr- 

(f, f^ FLr— 

j^l j. JT^rz^ Ft-T— 

OSL- ^- 

Fig.  33.     The  combination  5  and  8.     Compare  figure  32 


I06  UNIVERSITY  OF   MISSOURI  STUDIES 

Let  US  examine  the  movements  of  the  twenty-five  initial 
sections.  From  g  to  B  they  move  down,  and  from  B  to  D  up. 
From  D  to  F  down,  that  is,  from  an  ex- 
The  tone  8  treme  upward  position  to  a  medium  up- 

ward position ;  and  from  F  to  H  they 
move  up  again,  that  is.  from  a  medium  upward  position  to  an 
extreme  upward  position.  From  H  to  J  they  move  down,  and 
from  J  to  L  up,  completing  thus  the  third  down  and  up  move- 
ment. From  L  to  N  they  move  down  and  from  N  to  P  up, 
completing  thus  the  fourth  down  and  up  movement.  From 
P  to  R  they  move  down,  and  from  R  to  T  up,  completing 
thus  the  fifth  down  and  up  movement.  From  T  to  V  down, 
and  from  V  to  X  up,  completing  thus  the  sixth  down  and  up 
movement.  From  X  to  Z  down,  and  from  Z  to  ©  up,  com- 
pleting thus  the  seventh  down  and  up  movement.  From  $8 
to  S)  down  and  from  S)  to  S'  up  again.  The  nerve  ends  lo- 
cated on  these  twenty-five  sections  therefore  receive  eight 
shocks  during  the  period,  and  accordingly,  convey  the  sen- 
sation of  the  tone  8. 

The  thirty-six  sections  from  the  twenty-sixth  to  the  sixty- 
first  move  down  from  g  to  B.  and  up  from  B  to  D.  Down 
from  D  to  F,  and  up  from  F  to  H.  Down  from  H  to  J,  and 
up  from  J  to  L.  Down  from  L  to  X.  and  up  from  N  to  P. 
Down  from  P  to  R,  and  up  from  R  to  T.  Down  from  T  to 
V,  and  up  from  V  to  X.  Down  from  X  to  2),  and  up  from  S) 
to  5^.  The  nerve  ends  located  on  these  sections  therefore 
receive  seven  shocks  during  the  period.  But,  in  accordance 
with  previous  considerations,  it  is  highly  improbable  that  they 
could  convey  the  sensation  of  the  tone  7.  When  seven  shocks 
are  received  in  time  intervals  identical  with  those  of  the  tone 
8,  and  when  the  eighth  shock,  at  the  moment  Z,  chances  to 
be  omitted,  it  is  rather  to  be  expected  that  the  tone  8  is( 
heard,  only  with  a  little  pause  or,  perhaps,  merely  a  diminu- 
tion of  intensitv  at  the  moment  Z.     The  sensation  conveved 


MECHANICS  OF  THE  INNER  EAR  I07 

by  these  nerve  ends,  then,  is  probably   the   tone     8'    slightly 
beating,  that  is,  being  characterized  by  a  slight  roughness. 

The  nineteen  sections  from  the  sixty-second  to  the  eight- 
ieth move  down  from  g^  to  B,  and  up  from  B  to  D.  Down 
from  D  to  F,  and  up  from  F  to  H.  Dbwn  from  H  to  J,  and 
up  from  J  to  L.  Down  from  L  to  N,  and  up  from  N  to  P. 
Down  from  P  to  R,  and  up  from  R  tO'  T.  Down  from  T 
to  V,  and  up  from  V  to  X.  Down  from  X  to  2),  and  up  from 
S)  to  g.  The  nerve  ends  located  on  these  sections  therefore 
receive  seven  shocks  during  the  period ;  but,  here  as  above, 
it  is  highly  improbable  that  they  could  convey,  merely  be- 
cause of  the  omission  of  the  stimulus  at  Z,  the  sensation  of  the 
tone  7  instead  of  8.  Most  probably  the  tone  heard  is  8  with 
a  slight  roughness. 

The  fifty-eight  sections  from  the  eighty-first  to  the  one 
hundred    and    thirty-eighth    move   down   from  g  to  B,  and   up 

from  B  to  H.  Down  from  H  to  J,  and  up 
The  tone  6  from  J  to  L.     Down  from  L  to  N,  and  up 

from  N  to  P.  Down  from  P  to  R,  and 
up  from  R  to  T.  Down  from  T  to  V,  and  up  from  V  to  X. 
Down  from  X  to  S),  and  up  from  S)  to  g^.  The  nerve  ends 
located  on  these  sections  therefore  receive  six  shocks  during 
the  period. 

The  fifty-six  sections  from  the  one  hundred  and  thirty- 
ninth  to  the  one  hundred  and  ninety-fourth  move  down  from 

g  to  B,  and  up  from  B  to  H.  Down  from 
The  tone  5  H  to  J,  and  up  from  J  to  L.     Down  from 

L  to  R,  and  up  from  R  to  T.  Down  from 
T  to  V,  and  up  from  V  to  X.  Down  from  X  to  ®,  and  up 
from  ®  to  g.  The  nerve  ends  located  on  these  sections  there- 
fore receive  five  shocks  during  the  period. 

All  the  following  sections  to  the  three  hundred  and  ninety- 
first  move  down  and  up  five  times  during  the  period.  Let 
us  examine  only  the  last  twenty-five  of  this  group.     They  move 


I08  UNIVERSITY  OF  MISSOURI  STUDIES 

down  from  A  to  C  and  up  from  C  to  I.  Down  from  I 
to  K,  and  up  from  K  to  M.  Down  from  M  to  S,  and  up  from 
S  to  U.  Down  from  U  to  W,  and  up  from  W  to  Y.  Down 
from  Y  to  @,  and  up  from  (S  to  @=A. 

The  nine  sections  from  the  three  hundred  and  ninety- 
second  to  the  four  hundredth  move  down  from  Y  to  C,  and 
up  from  C  to  I.  Down  from  I  to  K,  and 
The  tone  3  up  from  K  to  M.     Down  from  M  to  W, 

and  up  from  W  to  Y.  The  nerve  ends  lo- 
cated on  these  sections  therefore  receive  three  shocks  during 
the  period. 

The  sections  from  the  four  hundred  and  first  to  the  four 
hundred  and  twenty-fourth  move  down  from  Y  to  C,  and  up 
from  C  to  M.  Down  from  M  to  W,  and 
The  tone  2  up  from  W  to  Y.    The  sections  from  the 

four  hundred  and  twenty-fifth  to  the  four 
hundred  and  ninety-eig-hth  move  down  from  A  to  C,  and  up 
from  C  to  M.  Down  from  M.  to  W,  and  up  from  W  to  @=A. 
The  sections  from  the  four  hundred  and  ninety-ninth  to  the 
five  hundred  and  forty-sixth  move  down  from  A  to  K,  and 
up  from  K  to  M.  Down  from  M  to  W,  and  up  from  W  to 
@=A.  The  sections  from  the  five  hundred  and  forty-seventh 
to  the  five  hundred  and  sixty-second  move  down  from  A  to 
K,  and  up  from  K  to  U.  Down  from  U  to  W,  and  up  from 
W  to  ®=A.  The  nerve  ends  located  on  these  sections  of 
the  partition  therefore  receive  two  shocks  during  the  period. 

The  sections  from  the  five  hundred  and  sixty-third  to  the 
five  hundred  and  seventy-ninth  move  down  from  U  to  K,  and 
up  from  K  to  U.  The  sections  from  the 
The  tone  1  five  hundred  and  eightieth  to  the  five  hun- 

dred and  ninety-sixth  move  down  from  A 
to  K,  and  up  from  K  to  @=A.  The  nerve  ends  located  on 
these  sections  therefore  receive  one  shock  during  the  period. 

The  relative  intensities  of  the  several  tones,  if  we  accept 


MECHANICS  OF  THE  INNER  EAR 


109 


the  third,  fourth,  and  sixth  provisional  assumptions,  are  shown 

in  the  following  table,  which  contains  the 

The  relative  number  of    partition  sections     conveying 

intensities  each  tone  in  absolute  numbers  as  well  as 

in  percentages. 


Tones 

8,   smooth 

8,  rough 

6 

5 

3 

2 

T 

Intensities 

Percent- 
ages 

25 
4-2 

55 
9.2 

58 
9-7 

253 
42.5 

9 
1-5 

162 

27.2 

34 
5-7 

Since  in  the  case  just  studied  the  higher  of  the  two  primary- 
tones,  though  weak,  is  yet  audible,  let  us  still  further  change  the 
relative  intensities  of  the  objective  tones  in  favor  of  the  lower  one. 
The  curve  in  figure  34  represents  the  function 
f(x)=:  3sin5;r  +  sinS.^. 

This  signifies  that  the  stirrup  move- 
ment eight  has  an  amplitude  which  is  only 
one-third  of  the  amplitude  of  the  stirrup 
movement  five.  The  table  below  contains 
all  the  abscissa  and  ordinate  values  of  the 
maxima  and  minima  and  of  the  inflection 
points  of  the  curve. 


The  combina- 
tion 5  and  8. 
Amplitude  of  8 
still  less 


no 


UNIVERSITY  OF  MISSOURI  STUDIES 


Interval  5:8,  Amplitudes  3:1 


Ordinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Inf. 

0 

0 

397 

V 

376 

Max. 

+  376 

149 

773 

w 

376 

Inf. 

+  114 

283 

511 

X 

262 

Min . 

—  219 

483 

178 

Y 

333 

Inf. 

Z 

Max. 

SI 

Inf. 

58 

Min. 

e 

Inf. 

+  140 

820 

.?37 

® 

359 

Max. 

+  336 

949 

733 

e 

196 

Inf. 

—  41 

1 106 

356 

& 

377 

Min. 

—  397 

1250 

0 

A 

356 

Inf. 

—  77 

1389 

320 

B 

320 

Max. 

+  281 

1558 

678 

C 

358 

Inf. 

D 

Min. 

E 

Inf. 

0 

1800 

397 

F 

281 

Max. 

G 

Inf. 

H 

Min. 

—  281 

2042 

116 

I 

281 

Inf. 

+  77 

2211 

474 

J 

358 

Max. 

+  397 

2350 

794 

K 

320 

Inf. 

+  41 

2494 

438 

L 

356 

Min. 

-336 

2651 

61 

M 

377 

Inf. 

—  140 

2780 

257 

N 

196 

Max. 

0 

Inf. 

P 

Min. 

Q 

Inf. 

R 

Max. 

+  219 

3"7 

616 

S 

359 

Inf. 

—  114 

3317 

283 

T 

333 

Min. 

-376 

3451 

21 

U 

262 

Inf. 

0 

3600 

397 

V 

376 

MECHANICS  OP  THE  INNER  EAR  1 II 

The  successive  positions  of  the  partition  corresponding, 
under  the  sixth  provisional  assumption,  to  the  maxima,  minima, 
and  inflection  points  of  the  curve  are  shown  in  figure  35. 

°§8888888 
B, i .— 

Cr f ^-' 

/•L i. -fr^rrrrx-, 

/l 1 1 1 

J^ ^ t,,,,,^^j:=.rrrra-, 

Kr T- 

/L 1 ,-. 

A/. fa^-- . 

^r  f -b^" 

Sr + b:zj=r- 

TL i p:3=r3rr=3t.^^- 

Ui. hr- 

V, ■^ 4a- 

W^ Fb— 

Y\ 4 -n-^-^ 

Ki .^-------na-- 

a, \- -fc=.^::,^^=zRi»  — 

0?. -pri- -FLr— 

4\ 4. -rrq. -nu- 
ts I ^ — 

^ig-  35-     The  combination  5  and  8.     Compare  figure  34 

The  five  hundred  and  fifty-five  initial  sections  of  the  par- 
tition move  down  and  up  five  times  during  the  period.     Let 
us  here  closely  examine  only  the  one  hun- 
The  tone  5  dred  and   ninety-six   initial    sections    and 

the  one  hundred  and  seventy-eight  most 
distant  sections  of  this    group.     The     initial     sections     move 


112  UNIVERSITY  OF  MISSOURI  STUDIES 


down  from  g  to  B,  and  up  from  B  to  F.  Down  from  F  to  J, 
and  up  from  J  to  L.  Down  from  L  to  N,  and  up  from  N  to 
T.  Down  from  T  to  V,  and  up  from  V  to  X.  Down  from  X 
to  2),  and  up  from  S)  to  g.  The  sections  from  the  three  hun- 
dred and  seventy-eighth  to  the  five  hundred  and  fifty-fifth 
move  down  from  A  to  C,  and  up  from  C  to  I.  Down  from 
I  to  K,  and  up  from  K  to  M.  Down  from  M  to  S,  and  up  from 
S  to  U.  Down  from  U  to  W,  and  up  from  W  to  Y.  Down 
from  Y  to  6  and  up  from  @  to  (S=:A.  The  nerve  ends  of  all 
these  sections  therefore  receive  five  shocks  during  the  period. 
The  seven  sections  from  the  five  hundred  and  fifty-sixth 
to  the  five  hundred  and  sixty-second  move  down  from  Y  to 
C,  and  up  from  C  to  I.  Down  from  I  to 
The  tone  3  K,  and  up  from  K  to  M.     Down  from  M 

to  W,  and  up  from  W  to  Y.  The  nerve 
ends  located  on  these  sections  therefore  receive  three  shocks 
during  the  period. 

The  sections  of  the  partition  from  the  five  hundred  and 
sixty-third  to  the  five  hundred  and  seventy-second  move  down 
and  up  twice  during  the  period.     Let  us 
The  tone  2  here  examine  only  the  sections  from  the 

five  hundred  and  sixty-third  to  the  five 
hundred  and  ninety-fifth.  They  move  down  from  Y  to  C,  and 
up  from  C  to  M.  Down  from  M  to  W,  and  up  from  W  to  Y. 
The  nerve  ends  located  on  these  sections  therefore  receive  two 
shocks  during  the  period. 

The  partition  sections  from  the  seven  hundred  and  fifty- 
third  to  the  seven  hundred  and  seventy-third  move  down  from 
U  to  K,  and  up  from  K  to  U.     The  sec- 
The  tone  1  tions  from  the  seven  hundred  and  seventy- 

fourth  to  the  seven  hundred  and  ninety- 
fourth  move  down  from  A  to  K,  and  up  from  K  to  @=A. 


MECHANICS  OF  THE  INNER  EAR 


"3 


All  the  nerve  ends  on  these  sections  therefore  receive  one 
shock  during  the  period. 

The  relative  intensities  of    the    several 
The  relative  tones  under  the  third,  fourth,  and  sixth  pro- 

intensities  visional  assumptions  are  shown  in  the  fol- 

lowing table. 


Tones 

S 

3 

2 

I 

Intensities 

Percent- 
ages 

555 
69.9 

7 
•9 

190 
23-9 

42 
5-3 

Having  studied  the  effect  of  changing  the  relative  intensities 
of  the  objective  tones  in  favor  of  the  lower  one,  we  shall  now  in- 
vestigate the  effect  of  increasing  the  intensity  of  the  higher  objec- 
tive tone.    The  curve  in  figure  36  represents  the  function 
f{x)  =  sin5;r  -|-  2sin8ir. 
The  stirrup  movement  eight  has  an  am- 
plitude which  is  twice  the  amplitude  of 
the  stirrup  movement  five.    The  table  be- 
low   contains    the    abscissa     and     ordinate 
values  of  the  maxima,  minima,  and  inflec- 
tion points. 


The  combination 
5  and  8.    The 
amplitude  of  8  is 
greater  than  of  5 


114 


UNIVERSITY  OF   MISSOURI  STUDIES 


Interval  5:8,  Amplitudes  1:2 


Ordinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Int. 

0 

0 

298 

V 

286 

Max. 

+  286 

123 

584 

W 

286 

Inf. 

+  55 

237 

353 

X 

231 

Min. 

—  190 

360 

108 

Y 

245 

Inf. 

—  49 

460 

249 

Z 

141 

Max. 

+  102 

568 

400 

SI 

151 

Inf. 

—  26 

669 

272 

93 

128 

Min. 

—  152 

767 

146 

S 

126 

Inf. 

+  60 

886 

358 

® 

212 

Max. 

+  262 

996 

560 

(£ 

202 

Inf. 

—  20 

1120 

278 

g 

282 

Min. 

—  298 

1241 

0 

A 

278 

Inf. 

—  40 

1359 

258 

B 

258 

Max. 

+  229 

1484 

527 

C 

269 

Inf. 

-I-  60 

1588 

358 

D 

169 

Min. 

—  120 

1702 

178 

E 

180 

Inf. 

0 

1800 

298 

F 

120 

Max. 

+  120 

1898 

418 

G 

120 

Inf. 

—  60 

2012 

238 

H 

180 

Min. 

—  229 

2H6 

69 

I 

169 

Inf. 

+  40 

2241 

3.38 

J 

269 

Max. 

+  398 

2359 

596 

K 

258 

Inf. 

+  20 

2480 

318 

L 

278 

Min. 

—  262 

2604 

36 

M 

282 

Inf. 

—  60 

2714 

238 

N 

202 

Max. 

+  I.S2 

2833 

450 

0 

212 

Inf. 

+   26 

2931 

324 

P 

126 

Min. 

—  102 

3032 

196 

Q 

128 

Inf. 

+  49 

3140 

347 

R 

151 

Max. 

+  190 

3240 

488 

S 

141 

Inf. 

—  55 

3363 

243 

T 

245 

Min. 

—  286 

3477 

12 

U 

231 

Inf. 

0 

3600 

298 

V 

286 

MECHANICS  OF  THE  INNER  EAR 


"5 


The  successive  positions  of  the  partition  corresponding, 
under  the  sixth  provisional  assumption,  to  the  maxima,  min- 
ima, and  inflection  points  of  the  curve  are  shown  in  figure  37. 

The  two  hundred  and  forty  initial  sections  move  down 
and  up  8  times  during  the  period.  Let  us  here  examine  only 
the  nine  most  distant  sections  of  this 
The  tone  8  group,  from  the  two  hundred  and  thirty- 

second  to  the  two  hundred  and  fortieth. 
They  move  down  from  A  to  B,  and  up  from  B  to  E.  Down 
from  E  to  G,  and  up  from  G  to  I.  D^own  from  I  to  J,  and  up 
from  J  to  L.  Down  from  L  to  O,  and  up  from  O  to  Q.  Down 
from  Q  to  S,  and  up  from  S  to  T.  Down  from  T  to  V,  and  up 
from  V  to  Y.  Down  from  Y  to  ST,  and  up  from  ST  to  ®.  Down 
from  (£  to  @,  and  up  from  ©  to  @=A.  The  nerve  ends  lo- 
cated on  these  sections  therefore  receive  eight  shocks  during 
the  period. 

The  fourteen  sections  from  the  two  hundred  and  forty- 
first  to  the  two  hundred  and  fifty-fourth  do  not  move  down 
from  E  to  G.  The  nerve  ends  located  on  these  sections  do  not, 
therefore,  receive  a  shock  between  E  and  I,  but  receive  the 
other  seven  shocks  in  the  same  manner  as  the  two  hundred  and 
forty  initial  sections.  For  the  same  reasons  as  in  the  similar 
cases  with  which  we  have  met  before,  it  is  not  probable  that 
these  nerve  ends  convey  the  tone  7,  but  rather  the  tone  8  with 
a  slight  beat  occurring  once  during  the  period,  producing  a 
slightly  rough  tone  8. 

The  sections  of  the  partition  from  the  twO'  hundred  and 
fifty-fifth  to  the  four  hundred  and  fifty-second  move  down  and 
up  five  times  during  the  period.  Let  us 
The  tone  5  examine  those  from  the  two  hundred  and 

fifty-fifth  to  the  two  hundred  and  fifty- 
eighth.  They  move  down  from  A  to  B,  and  up  from  B  to  E. 
Down  from  E  to  J,  and  up  from  J  to  L.  Down  from  L  to  O, 
and  up  from  O  to  U.     Down  from  U  to  V,  and  up  from  V 


H6  UNIVERSITY  OF  MISSOURI  STUDIES 

O  O  Q  O  Q  O  O 

o  o  o  o  o  o 

M  cj  to  ^-  lo  so 

>JL J.^ 

r ^ -I-- 

g, «■ 1— 

L X -rr^zTra.- 

^ .^ ,  — .rrr^ ^r^^^m.. 

L 1 ^. ---1 4- -L- 

/> 4 4 u- 

^ ^ ■ ^.^^..^j^rr^^.- 

/Cp .--,-. 

L 1 _ 

Mi ,,_-.-„„.. -,,,-1--^,, 

, ^ ,,„ ,,, _,^^--U^,. 

L, !■■■ ,_ ^^_ 

, ^ ^ 4. b^^— 

S^ ^ ,.___. 

L X . _._, ,.._ 

U^ ^. 

^ ^ )^.. 

Wr ^.- 

L 1. ^.. 

Ku ^ ^. 

^ -^ ^ ^^_ 

Ai h 1 R-, — 

I 4 1 1 n-y— 

, F^lizid ^ PI-,— 

(g, jT^n -n-^— 

I 4 .pr^ .n.^,. 

61 

Fig.  37.     The  combination  5  and  8.     Compare  figure  36 


MECHANICS  OF  THE  INNER  EAR 


117 


to  Y.  Down  from  Y  to  31,  and  up  from  St  to  @=A.  The 
nerve  ends  located  on  these  sections  therefore  receive  five 
shocks  during  the  period. 

The   sections  from  the  four  hundred   and  fifty-third  to 
the  four  hundred  and  fifty-eighth  move  down  from  Y  to  C, 
and  up  from  C  to  I.     Down  from  I  to  K, 
The  tone  3  and  up  from  K  to  M.     Down  from  M  to 

W,  and  up  from  W  to  Y.  The  nerve  ends 
located  on  these  sections  therefore  receive  three  shocks  dur- 
ing the  period. 

The  sections  of  the  partition  from  the  four  hundred  and 
fifty-ninth  to  the  five  hundred  and  seventy-second  move  down 
and  up  twice  during  the  period.     Let  us 
The  tone  2  examine,   for   example,   the   four  hundred 

and  fifty-ninth  and  the  four  hundred  and 
sixtieth.  They  move  down  from  Y  to  C,  and  up  from  C  to 
M.  Down  from  M  to  W,  and  up  from  W  to  Y,  The  nerve 
ends  located  on  these  sections  therefore  receive  two  shocks 
during  the  period. 

The  sections  of  the  partition  from  the  five  hundred  and 
seventy-third  to  the  five  hundred  and  eighty-fourth  move  down 
from  U  to  K,  and  up  from  K  to  U.    The 
The  tone  1  sections  from  the  five  hundred  and  eighty- 

fifth  to  the  five  hundred  and  ninety-sixth 
mjove  down  from  A  to  K,  and  up  from  K  to  A.  The  nerve 
ends  located  on  these  sections  therefore  receive  one  shock 
during  the  period. 

The   relative   intensities   of   the   several 
The  relative  tones  under  the  third,  fourth,  and  sixth  pro- 

intensities  visional  assumptions  are  shown  in  the  fol- 

lowing table. 


Tones 

8  smooth 

8  rough 

5 

3 

2 

I 

Intensities 
Percent- 
ages .    . . 

240 
40-3 

14 
2.4 

198 
33-3 

6 

I.O 

114 
19. 1 

24 
4.0 

Il8  UNIVERSITY  OF  MISSOURI  STUDIES 

The  curve  in  figure  38  represents  the  function 

f(x)=  sin5:jr  -|-  3sin8;ir. 

_,,  ,  .       .         The  stirrup  movement  eiffht  has  an  ampli- 

Irie  combination  . 

5  and  8      The  tude  three  times  as  great  as  that  of  five. 

amplitude  of  8  The  table  below  contains  the  abscissa  and 

is  three  times  ordinate   values   of  the   maxima,   minima, 

that  of  5  and  inflection  points. 

The  successive  positions  of  the  partition  corresponding, 
under  the  sixth  provisional  assumption,  to  the  maxima,  min- 
ima, and  inflection  points  of  the  curve  are  shown  in  figure  39. 

The  four  hundred  and  thirty-eight  initial  sections  of  the 
partition  move  down  and  up  eight  times  during  the  period. 
Let  us  examine  those  from  the  three  hun- 
The  tone  8  dred  and  eighty-sixth  to  the  four  hundred 

and  thirty-eighth.  They  move  down  from 
A  to  C  and  up  from  C  to  E.  Down  from  E  to  G,  and  up  from 
G  to  I.  Down  from  I  to  K  and  up  from  K  to  M.  Down  from 
M  to  O,  and  up  from  O  to  Q.  Down  from  Q  to  S,  and  up 
from  S  to  U.  Down  from  U  to  W,  and  up  from  W  to  Y. 
Down  from  Y  to  %,  and  up  from  St  to  d.  Down  from  ®  to  (S, 
and  up  from  @  to  @=:A.  The  nerve  ends  located  on  these  sec- 
tions therefore  receive  eight  shocks  during  the  period. 

The  sections  from  the  four  hundred  and  thirty-ninth  to 
the  four  hundred  and  fifty-first  move  down  and  up  only  seven 
times,  since  they  do  not  make  a  double  movement  between 
E  and  I.  In  accordance  with  our  former  considerations,  how- 
ever, in  similar  cases,  it  does  not  seem  probable  that  the  nerve 
ends  located  on  these  sections  should  convey  any  other  tone 
than  the  tone  8  of  a  slight  roughness. 

The  sections  of  the  partition  from  the  four  hundred  and 

fifty-second  to  the  six  hundred  and  forty-seventh  move  down 

five     times      during     the     period.        Let 

The  tone  5  us  examine  those  from  the  four  hundred 

and  fifty-second  to  the  four  hundred  and 

eighty-ninth.     They  move  down  from  A  to  C,  and  up  from 


MECHANICS  OF  THE  INNER  EAR 


IK 


Interval  5:8,  Amplitudes  1:3 

Ordinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Inf. 

0 

0 

398 

V 

385 

Max. 

+  385 

120 

783 

W 

385 

Inf. 

+  56 

233 

454 

X 

329 

Min. 

—  287 

353 

III 

Y 

343 

Inf. 

-  46 

457 

352 

Z 

241 

Max. 

+  202 

566 

600 

2t 

248 

Inf. 

—  25 

671 

373 

S3 

227 

Min. 

—  249 

774 

149 

e 

224 

Inf. 

+  .58 

890 

456 

® 

307 

Max. 

+  360 

lOOI 

758 

g 

302 

Inf. 

—  18 

1121 

380 

g 

378 

Min. 

-  398 

1240 

0 

A 

380 

Inf. 

—  42 

1356 

356 

B 

356 

Max. 

+  326 

1477 

724 

C 

368 

Inf. 

+  58 

1584 

456 

D 

268 

Min. 

—  219 

1697 

179 

E 

277 

Inf. 

0 

1800 

398 

F 

219 

Max. 

+  219 

1903 

617 

G 

219 

Inf. 

-  58 

2016 

340 

H 

277 

Min. 

—  326 

2123 

72 

I 

268 

Inf. 

+  42 

2244 

440 

J 

368 

Max. 

+  398 

2360 

796 

K 

.  356 

Inf. 

+  18 

2479 

416 

L 

380 

Min. 

—  360 

2599 

38 

M 

378 

Inf. 

-  58 

2710 

340 

N 

302 

Max. 

+  249 

2826 

647 

0 

307 

Inf. 

+  25 

2929 

423 

P 

224 

Min. 

—  202 

3034 

196 

Q 

227 

Inf. 

+  46 

3143 

444 

R 

248 

Max. 

+  287 

3247 

68s 

S 

241 

Inf. 

-  56 

3367 

342 

T 

343 

Min. 

-38.'; 

3480 

13 

U 

329 

Inf. 

0 

3600 

398 

V 

385 

I20 


MECHANICS  OF  THE  INNER  EAR  121 

C  to  E.  Down  from  E  to  K,  and  up  from  to  K  to  M.  Down 
from  M  to  O,  and  up  from  O  to  U.  Down  from  U  to  W,  and 
up  from  W  to  Y,  Dbwn  from  Y  to  %  and  up  from  %  to 
®=A.  The  nerve  ends  located  on  these  sections  therefore 
receive  five  shocks  during  the  period. 

The  five  sections  from  the  six  hundred  and  forty-eighth 
to  the  six  hundred  and  fifty-second  move  down  from  Y  to  C, 

and  up  from  C  to  I.  Down  from  I  to  K, 
The  tone  3  and  up  from  K  to  M.    Down  from  M  to  W, 

and  up  from  W  to  Y.  The  nerve  ends  lo- 
cated on  these  sections  therefore  receive  three  shocks  during 
the  period. 

The  sections  of  the  partition  from  the  six  hundred  and 
fifty-third  to  the  seven  hundred  and  seventieth  move  down 

and  up  twice  during  the  period.  Let  us 
The  tone  2  examine  those  from  the  six  hundred  and 

fifty-third  to  the  six  hundred  and  seventy- 
second.  They  move  down  from  Y  to  C,  and  up  from  C  to  M. 
Down  from  M  to  W,  and  up  from  W  to  Y.  The  nerve  ends 
located  on  these  sections  therefore  receive  two  shocks  during 
the  period. 

The  sections  from  the  seven  hundred  and  seventy-first 
to  the  seven  hundred  and  eighty-third  move  down  from  U 

to  K,  and  up  from  K  to  U.  The  sections 
The  tone  1  from  the  seven  hundred  and  eighty-fourth 

to  the  seven  hundred  and  ninety-sixth  move 
down  from  A  to  K,  and  up  from  K  to  ®=A.  The  nerve 
ends  located  on  these  sections  therefore  receive  one  shock  during 

the  period. 

The  relative  intensities    of    the    several 
The  relative  ^nes  under  the  third,  fourth,  and  sixth  pro- 

intensities  visional  assumptions  are  shown  in  the  fol- 

lowing table : 


122 


UNIVERSITY  OF  MISSOURI  STUDIES 


Tones 

8  smooth 

8  rough 

5 

3 

2 

I 

Intensities 

Percent- 
ages 

438 
S5-0 

13 
1.6 

196 
24.6 

5 
.6 

118 
14.8 

26 
3-3 

It  is  interesting'  to  compare  the  intensities  of  the  several 
tones  in  the  last  five  cases,  all  representing  the  combination  8  plus 

5  of  stirrup  movement,  but  differing  in  the 
Comparison  of  relative  amplitudes  of  8  and  5.  The  table  con- 

the  last  live  ^^^^^  ^^^  percentages  of  the  five  preceding  ta- 

cases 

bles.  The  first  two  columns  show  the  ra- 
tio of  the  amplitudes  of  the  stirrup  movements  of  8  and  5 
For  example,  in  the  first  case  this  ratio  is  as  3:1  or  seventy- 
five  to  twenty-five ;  in  the  fifth  case  as  1:3  or  twenty-five  to 
seventy-five.  The  columns  to  the  right  contain  the  relative 
intensities  of  the  several  tones  calculated  under  the  provisional 
assumptions. 


Amplitudes 
of  stirrup 
movement 

Subjective  (theoretic)  intensity 

8 

5 

8 

6 

5 

3 

2 

I 

75 
67 
50 
33 
25 

25 
33 
50 
67 

75 

56.6 
42.7 
16. 1 

13-4 

3-0 
9-7 

24.6 
33-2 
48.0 

42.5 
69.9 

.6 

I.O 

1.8 

1-5 

•9 

14.8 
19. 1 
25.6 
27.2 
23-9 

3-3 
4.0 

5-5 
5-7 
5-3 

We  notice  that  the  tone  8  decreases  in  intensity  from  56.6 
to  42.7,  to  16.1,  to  13.4,  and  finally  disappears  entirely.  This 
latter  case,  however,  does  not  mean  that  now  the  tone  5  is 


MECHANICS  OF  THE  INNER  EAR 


123 


alone  audible.  We  see  from  the  table  that  even  now,  in  ad- 
dition to  5,  the  very  w^eak  difference  tone  3  and  the  fairly 
strong  difference  tones  2  and  1  are  to  be  expected  by  the  ob- 
server.^ 

As  to  the  several  dift'erence  tones,  the  most  favorable  con- 
dition for  6  seems  to  be,  to  have  the  component  5  of  the  com- 
pound stirrup  movement  somewhat  more  pronounced  than  8. 
It  appears,  however,  that  in  no  case  will  this  difference  tone 
become  very  conspicuous.  The  most  favorable  condition  for 
the  difference  tone  3  seems  to  be,  to  have  the  component  8  of 
stirrup  movement  about  as  strong  as  5.  The  difference  tones 
2  and  1,  on  the  other  hand,  appear  with  a  maximum  of  in- 
tensity when  the  component  5  of  stirrup  movement  is  some- 
what greater  than  8.  But  their  intensities  are  but  little  less 
in  case  the  amplitudes  of  the  two  stirrup  movements  8  and 
5  are  equal.  With  respect  to  all  the  difference  tones  taken  to- 
gether, it  appears  that  these  tones  are  very  unfavorably  influ- 
enced by  a  considerable  difference  in  the  amplitudes  of  the 
component  stirrup  movements,  for  no  difference  tone  has  a 
maximum  intensity  in  either  the  first  or  the  fifth  case.      And 


^Although  this  booklet  is  devoted  to  theory  and  not  to  experimental 
methods  of  research,  I  cannot  refrain  from  mentioning  a  way  of  testing 
the  theoretical  results  just  spoken  of,  because  it  is  so  easy  for  anj'  one  who 
possesses  a  skillful  hand  and  a  trained  ear,  and  the  observation  to  be  made 
is  so  pretty.  No  instruments  are  required  but  two  good  tuning  forks  on 
resonance  boxes,  accurately  tuned  in  the  ratio  of  5:8,  and  a  bass  bow.  The 
fork  5  must  be  sounded  first,  as  strongly  as  possible,  and  it  is  necessary  to 
have  a  fork  which  continues  to  sound  strongly  for  quite  a  while.  Then 
the  bow  is  applied  with  the  most  delicate  touch  to  the  fork  8.  It  is  neces- 
sary for  the  success  of  the  experiment  that  the  intensity  of  the  higher  tone 
vibration  be  increased  from  zero  very  slowly  and  uniformly.  If  these  con- 
ditions are  fulfilled,  one  suddenly  hears  the  low  difference  tones  i  and  2 
being  added  distinctly  to  the  tone  5,  whereas  of  8  no  trace  is  yet  audible. 
If  now  the  fork  8  is  left  to  itself,  and  the  fork  5  is  stopped  by  firmly  touch- 
ing it  with  a  finger,  the  tone  5  together  with  the  difference  tones  disap- 
pears, but  immediately  one  hears  with  surprising  clearness  the  tone  8,  which 
a  moment  ago  was  entirely  inaudible.  No  similar  observation  can  be 
made  with  a  strongly  sounding  fork  8  and  a  weakly  sounding  fork  5.  Ac- 
cording to  our  theoretic  deduction  the  lower  tone  does  not  become 
inaudible  when  the  amplitude  of  8  is  three  times  that  of  5,  but  still  has  a 
respectable  intensity. 


124 


UNIVKRSITY  OF  MISSOURI  STUDIES 


a  prevailing  intensity  of  8  seems  to  be  even  less  favorable  to  the 
difference  tones  than  a  prevailing  intensity  of  5.  All  these  con- 
clusions have,  of  course,  only  a  relative  value,  since  taking  into 
account  the  various  provisional  assumptions  changes  the  result 
considerably. 

Let  us  study  one  more  combination  of  sinusoidal  stirrup 
movements.    We  have  had  only  one  interval  greater  than  an 
octave,    the    combination    4    and    9.       But 
we  did  not,  then,  take  into  account  the  in- 
flection points  of   the   curve.     Let    us    do 
this  with  the  combination  3  and  8,  taking 
the  amplitude  of  3  twice  as  great  as  that 
of  8'.    This  ratio  of  the  amplitudes  is  arbi- 
trarily chosen.     But  the  selection  of  equal 
amplitudes  would  be  no  less  arbitrary.     The  curve  in  figure 
40  represents  the  function 

f(x)  =  2sin3.«'  +  sin8;ir. 
The  table  below  contains  the  abscissa  and  ordinate  values  of 
the  maxima,  minima,  and  inflection  points  of  the  curve. 


The  combination 
3  and  8.     The 
amplitude  of  3 
twice  that  of  8 


Fig.  40.     The  combination  3  and  8 


The  successive  positions  of  the  partition  corresponding 
to  the  maxima,  minima,  and  inflection  points  are  shown  in 
figure  41. 

The  thirteen  initial  sections  of  the  partition  move  down 
from  g  to  B,  and  up  from  B  to  D.     Down  from  D  to  F,  and 


MECHANICS  OF  THE  INNER  EAR 


125 


Interval  3 : 8,  Amplitudes  2 :  i 


Ordinate 

Abscissa 

Ordinate 

Ordinate 
Difference 

Inf. 

0 

0 

297 

N 

228 

Max. 

+  228 

152 

525 

0 

238 

Inf. 

4  164 

245 

461 

P 

64 

Min. 

+  95 

353 

392 

Q 

69 

Inf. 

+  131 

435 

428 

R 

36 

Max. 

+  165 

510 

462 

S 

34 

Inf. 

—  60 

668 

237 

T 

225 

Min. 

—  273 

812 

24 

U 

213 

Inf. 

—  171 

920 

126 

V 

102 

Max. 

—  55 

1054 

242 

W 

116 

Inf. 

—  73 

1117 

224 

X 

18 

Min. 

—  90 

1177 

207 

Y 

17 

Inf. 

-f  1x3 

1337 

410 

Z 

303 

Max. 

+  297 

147 1 

594 

21 

184 

Inf. 

+  152 

1593 

449 

33 

145 

Min. 

—  13 

1745 

284 

e 

16s 

Inf. 

0 

1800 

297 

® 

13 

Max. 

+  13 

1855 

310 

(S 

13 

Inf. 

—  152 

2007 

145 

& 

165 

Min. 

—  297 

2129 

0 

A 

145 

Inf. 

—  113 

2263 

184 

B 

184 

Max. 

+  90 

2423 

387 

C 

203 

Inf. 

+  73 

2483 

370 

D 

17 

Min. 

+  55 

2546 

352 

E 

18 

Inf. 

+  171 

2680 

468 

F 

116 

Max. 

+  273 

2788 

570 

G 

102 

Inf. 

+  60 

2932 

357 

H 

213 

Min. 

-  16s 

3090 

132 

I 

225 

Inf. 

—  131 

3165 

166 

J 

34 

Max 

—  95 

3247 

202 

K 

36 

Inf. 

—  164 

3355 

133 

L 

69 

Min 

—  228 

3448 

69 

M 

64 

Inf. 

0 

3600 

297 

N 

228 

126  UNIVERSITY^  OF  MISSOURI  STUDIES 

up  from  F  to  H.     Down  from  H  to  J,  and  up  from  J  to  L. 

Down  from  L  to  N,  and  up  from  N  to  P. 
The  tone  8  Down  from  P  to  R,  and  up  from  R  to  T. 

Down  from^  T  to  V,  and  up  from  V 
to  X.  Down  from  X  to  Z,  and  up  from  Z  to  93.  Down  from 
93  to  3),  and  up  from  S?  to  ^5-  The  nerve  ends  located  on 
these  sections  therefore  receive  eight  shocks  during  the  pe- 
riod. 

Let  us  examine  the  sections  from  the  sixty-fifth  to  the 
sixty-ninth.  They  move  down  from  g  to  B,  and  up  from 
B  to  E.  Down  from  E  to  F,  and  up  from  F  to  H.  Down  from 
H  to  K,  and  up  from  K  to  L.  Down  from  L  to  N,  and  up  from 
N  to  Q.  Down  from  Q  to  S,  and  up  from  S  to  T.  Down  from 
T  to  V,  and  up  from  V  to  Y.  Down  fom  Y  to  Z,  and  up  from 
Z  to  93.  Down  from  93  to  ®,  and  up  from  ©  to  g.  The  nerve 
ends  located  on  these  sections  therefore  receive  eight  shocks 
during  the  period. 

The  seventieth  section  moves  down  from  ^  to  B,  and  up 
from  B  to  E.  Down  from  E  to  F,  and  up  from  F  to  H.  Down 
from  H  to  K,  and  up  from  K  to  M.  Down  from  M  to  N,  and 
up  from  N  to  Q.  Down  from  O^  to  S,  and  up  from  S  to  T.. 
Down  from  T  to  V,  and  up  from  V  to  Y.  Down  from  Y  to  Z, 
and  up  from  Z  to  93.  Down  from  $8  to  ©,  and  up  from  ©  to  ^. 
The  nerve  ends  located  on  this  section  therefore  receive  eight 
shocks  during  the  period. 

The  sections  of  the  partition  from  the  seventy-first  to  the 
one  hundred  and  second  move  down  from  g^  to  B,  and  up  from 

B  to  E.  Down  from  E  to  F,  and  up  from 
The  tone  6  F  to  H.    Down  from  H  to  N,  and  up  from 

N    to    T.    Down  from  T  to    V,    and    up 

from  V  to  Y.     Down  from  Y  to  Z,  and  up  from  Z  to  93.     Down 

from  $8  to  @,  and  up  from  ©  to  'Q.     The  nerve  ends  located 

on  these  sections  therefore  receive  six  shocks  during  the  period. 

The  sections  from  the  one  hundred  and  third  to  the  one 


1 28  UNIVERSITY  OF  MISSOURI  STUDIES 

hundred  and  forty-fifth  move  down  from  g  to  B,  and  up  from 
B  to  E.    Down  from  E  to  F,  and  up  from 
The  tone  5  F  to  H.    Down  from  H  to  N,  and  up  from 

N  to  T.  Down  from  T  to  Z,  and  up  from 
Z  to  35.  Down  from  $8  to  (S,  and  up  from  @  to  3^.  The  nerve 
ends  located  on  these  sections  therefore  receive  five  shocks 
during  the  period. 

The  sections  from  the  one  hundred  and  forty-sixth  to  the 
one  hundred  and  eighty-fourth  move  down  from  g  to  B,  and 
up  from  B  to  E.     Down  from  E  to  F,  and 
The  tone  4  up  from  F  to  H.     Down  from  H  to  N,  and 

up  from  N  to  T.  Down  from  T  to  Z,  and 
up  from  Z  to  3'-  The  nerve  ends  located  on  these  sections 
therefore  receive  four  shocks  during  the  period. 

The  sections  from  the  one  hundred  and  eighty-fifth  to  the 
four  hundred  and  fifty-sixth  move  down  and  up  three  times  dur- 
ing the  period.    Let  us  examine  those  from 
The  tone  3  the  one  hundred  and  eighty-fifth  to     the 

two  hundred  and  thirteenth.  They  move 
down  from  g  to  F,  and  up  from  F  to  H.  Down  from  H  to  N, 
and  up  from  N  to  T.  Down  from  T  to  Z,  and  up  from  Z  to 
g'.  The  nerve  ends  located  on  these  sections  therefore  receive 
three  shocks  during  the  period. 

The  sections  from  the  four  hundred  and  fifty-seventh  to 
the  four  hundred  and  sixty-eighth  move  down  from  A  to  F,  and 
up  from  F  to  M.    Down  from  M  to  ST,  and 
The  tone  2  ^P  from  2t  to  @=A.     The  sections     from 

the  four  hundred  and  sixty-ninth  to  the 
five  hundred  and  first  move  down  from  A  to  G,  and  up  from 
G  to  M.  Down  from  M  to  ST,  and  up  from  ST  to  @=A.  The 
sections  from  the  five  hundred  and  second  to  the  five  hundred 
and  forty-sixth  move  down  from  A  to  G,  and  up  from  G  to  U. 
Down  from  U  to  5[  and  up  from  21  to  @=A.  The  nerve  ends 
located  on  these  sections  therefore  receive  two  shocks  during 
the  period. 


MECHANICS  OF  THE  INNER  EAR 


129 


The  sections  of  the  partition  from  the  five  hundred  and 
forty-seventh  to  the  five  hundred  and  seventieth  move  down 
from  A  to  G,  and  up  from  G  to  ®=A.  The 
The  tone  1  sections  from  the  five  hundred  and  seventy- 

first  to  the  five  hundred  and  ninety-fourth 
move  down  from  A  to  ST,  and  up  from  21  to  @=A.  The  nerve 
ends  located  on  these  sections  therefore  receive  one  shock 
during  the  period. 

The  relative  intensities    of    the    several 
Th€  relative  tones  under  the  third,  fourth,  and  sixth  pro- 

intensities  visional  assumptions   are   shown  in  the   fol- 

lowing table: 


Tones 

8 

6 

5 

4 

0 
0 

2 

I 

Intensities 

Percent- 
ages 

70 
II. 8 

32 
5-4 

43 
7.2 

39 
6.6 

272 
45-S 

90 

15 -I 

48 
8.1 

We  notice  that  the  tone  3  is  theoretically  by  far  the 
strongest,  as  is  to  be  expected.  Of  the  difference  tones,  the 
tones  2,  1,  and  5  appear  to  be  somewhat  more  pronounced  than 
4  and  6.  Under  different  assumptions  concerning  the  physical 
properties  of  the  partition  these  results  would,  of  course,  be 
somewhat  different. 

Throughout  our  previous  discussions  we  have  never  taken 
into   account   the   possibility   of   the    tone   intensities     being 
further  modified  by  a  more  central  nervous 
Weber's  law  condition    like  the  one    usually    referred    to 

in  audition  as  Weber's  law.    All  our  various   approxi- 

mations towards  the  intensities  of  the  ner- 
vous processes  take  into  consideration  only  conditions  in  the 
peripheral  organ.  Whether  the  intensities  thus  found  are 
modified  more  centrally  in  accordance  with  Weber's  law  or 


130  UNIVERSITY  OF  MISSOURI  STUDIES 

not,  is  a  question  which  at  present  must  be  left  entirely  open, 
like  SO  many  others,  because  of  lack  of  experimental  data. 

Whenever  we  have     spoken  of  "amplitudes"     we  have 
meant  exclusively  the  amplitudes  of  stirrup     movement.     In 
order  to  make  use  of  our  theory  in  experi- 
Sounding  bodies      "cental  investigations  we  must  remember 
and  stirrup  ^^^  ^^ct  that  the  stirrup  movements  result 

movement  from  movements  of  the  tympanum,  trans- 

mitted by  a  rather  complicated  system  of 
levers,  the  auditory  ossicles.  It  is  quite  probable  that  the 
vibratory  movements  of  the  stirrup — even  when  these  move- 
ments are  highly  complex — are  approximately  like  those  re- 
ceived by  the  hammer,  the  ossicle  attached  to  the  tympanum. 
But  no  one  knows  as  yet  how  close  or  remote  this  approxima- 
tion is.  We  certainly  have  no  right  to  regard  this  approxima- 
tion as  infinitely  close,  save  by  way  of  a  provisional  assump- 
tion. 

The  movements  of  the  tympanum  result  from  rhythmical 
changes  of  the  density  of  the  external  air.  These  density 
changes,  in  experimental  investigations,  are  sometimes  pro- 
duced by  the  vibrations  of  gaseous  bodies,  as  in  labial  organ  pipes ; 
more  frequently,  however,  by  the  vibrations  of  solid  bodies,  par- 
ticularly of  tuning  forks  on  resonance  boxes.  Now,  we  must  not 
think  that  by  graphically  recording — which  is  a  comparatively 
easy  method — the  vibrations  of  a  tuning  fork,  we  obtain  a  record 
of  the  exact  form  of  the  resulting  air  waves.  It  has  been 
experimentally  and  mathematically  proved  that  the  form  of 
the  resulting  air  waves  must  be  more  or  less  different  from 
the  form  of  the  vibratory  movement  of  the  fork  or  other  solid 
body.  The  cause  of  this  alteration  of  the  form  is  to  be  found 
in  the  fact  that  the  layer  of  air  which  adjoins  the  solid  body 
and  therefore  directly  receives  the  impulses  from  that  body, 
is  unsymmetric  with  respect  to  its  elastic  properties,  because 


MECHANICS  OF  THE  INNER  EAR 


131 


it  is  in  contact  on  one  side  with  a  practically  unyielding  body, 
on  the  opposite  side  with  the  easily  yielding  air. 

It  is  of  the  utmost  importance,  therefore,  if  we  wish  to 
develop  the  theory  by  experimental  investigation,  to  keep 
free  from  the  delusion  that  any  of  the  above  theoretic  results, 
say,  in  the  case  of  the  combination  5  and  8  with  equal  ampli- 
tudes, applies  to  what  we  hear  in  case  two  tuning  forks  of  the 
vibration  ratio  5 : 8,  standing  at  an  arbitrary  distance  from 
our  ears  and  from  the  reflecting  walls  of  our  laboratory,  vi- 
brate with  equal  amplitudes.  It  is  only  by  way  of  approxima- 
tion that  we  can  derive  any  theoretic  conclusion  from  such  an 
experiment.  The  starting  point  of  our  theory  is  the  form  of 
movement  of  the  stirrup,  not  of  external  sounding  bodies. 

Under  ordinary  conditions,  it  is  a  great  advantage  that  we 
possess  two  organs  of  hearing,  some  distance  apart.  In  ex- 
perimental investigations,  however,  for  the 
The  duality  of  development  of  a  theory  of  audition,  this 
our  auditory  fact  is  often  a  serious  obstacle.    Since  we 

organ  cannot  make  experiments  on  audition  while 

soaring  like  an  eagle,  any  source  of  sound 
is  likely  to  surround  our  body  with  standing  waves,  resulting 
from  reflection.  Let  us  regard  the  velocity  of  sound  as  three 
hundred  and  thirty  meters,  the  distance  between  our  ears  as 
about  fifteen  centimeters.  A  tone  of  five  hundred  and  fifty 
complete  vibrations,  that  is,  a  tone  representing  the  ordi- 
nary human  voice  quite  well,  has  therefore  a  wave  length  of 
about  sixty  centimeters.  The  distance  between  a  nodal  point, 
where  the  rhythmic  density  changes  of  the  air  occur  with  full 
intensity,  and  a  point  of  maximum  vibratory  movement,  where 
there  are  practically  no  density  changes  affecting  the  tympa- 
num, is  then  about  fifteen  centimeters.  That  is,  it  might 
happen  with  standing  waves — if  the  head  was  kept  perfectly 
still — that  the  amplitude  of  one  of  the  components  of  stirrup 
movement  would  be  almost  zero  in  one  ear,  but  very  large  in 


132  UNIVERSITY  OF  MISSOURI  STUDIES 

the  other,  and  every  movement  of  the  head  would  greatly  alter 
these  conditions;  while  the  resulting  consciousness  would  be, 
of  course,  the  sum  total  of  the  tones  heard  by  each  ear.  It 
is  unnecessary  to  point  out  in  further  detail  how  this  fact  of 
hearing  with  two  ears  complicates  the  comparison  of  experi- 
mental results  with  the  theoretical  deductions  of  the  present 
study,  which  refer  only  to  one  stirrup  and  one  inner  ear,  and 
to  an  unalterable  form^  of  the  components  of  stirrup  movement 
in  a  given  case. 

The  fact  that  we  have  two  ears  would  be  irrelevant  only 
with  exceedingly  high  tones,  whose  wave  lengths  in  air  would 
be  so  small  as  to  be  negligible  quantities  in  comparison  with 
the  distance  between  our  ears,  as  the  wave  lengths  of  light 
are  negligible  quantities  in  comparison  with  the  distance  be- 
tween our  eyes  and  even  with  the  sensory  elements  of  each 
eye. 

Every  one  is  familiar  with  the  comparative  clearness  with 
which  the  ticking  of  a  watch  or  the  sound  of  a  tuning  fork 
is  perceived  if  the  vibrating  body  is  firmly 
Hearing  without  pressed  on  the  head  or  against  the  teeth. 
the  ear  drum  Some  believe    that    the  physiological    func- 

tion of  the  ear  in  such  a  case  is  not  essen- 
tially different  from  hearing  under  ordinary  conditions;  that  the 
sound  waves,  the  rhythmic  changes  of  molecular  density,  which 
pass  through  the  head,  naturally  pass  also  through  the  cavities 
of  the  head,  of  which  one,  the  middle  ear,  particularly  concerns 
us  here.  As  soon  as  rhythmic  changes  of  density  occur  in 
the  air  of  the  middle  ear,  the  tympanum  adjusts  itself  to  them 
by  rhythmically  moving  back  and  forth.  The  stirrup  cannot 
help  following  the  tympanum,  and  so  on.  The  only  difiference 
between  this  case  and  a  case  of  ordinary  hearing  consists 
in  the  fact  that  the  changes  of  density  of  the  air  affecting 
the  tympanum  originate  on  the  inside  of  the  tympanum  in- 


MECHANICS  OF  THE  INNER  EAR  1 33 

stead  of  on  the  outside,  and  that  they  must,  on  the  whole, 
be  much  weaker  in  the  former  case. 

There  can  be  Httle  doubt  that  the  process  just  spoken 
of  actually  occurs.  Some  have  insisted  also  on  the  possibility 
of  hearing  when  the  middle  ear  is  destroyed  and  no  movements 
of  the  stirrup  occur.  There  is  no  reason  why  we  should  a 
priori  deny  the  possibility  of  a  shock  being  received  by  the 
nerve  ends  whenever  a  rhythmical  change  of  molecular  den- 
sity takes  its  path  directly  through  them.  Such  a  molecular 
wave  might  originate  from  a  vibrating  solid  body  being  pressed 
against  skull  or  teeth,  or  from  sound  waves  in  the  air  strik- 
ing the  head  and  passing  through  it. 

We  must  not  overlook  the  fact,  however,  that  even  when 
the  tympanum  is  totally  destroyed,  if  sounds  are  perceived, 
the  perception  need  not  be  the  result  of  the  sound  waves 
simply  passing  through  the  nerves.  Even  in  such  a  case  stirrup 
movements  are  not  excluded.  If  we  blow  over  the  mouth  of 
a  bottle,  we  cause  rhythmical  changes  of  density  within  the 
bottle,  and,  as  a  natural  consequence,  the  air  in  the 
neck  of  the  bottle  rushes  back  and  forth.  These  move- 
ments may  often  be  observed  with  the  naked  eye  when  a  fiber 
adherent  to  the  inside  of  the  neck  of  a  bottle  is  forced  by  friction 
to  follow  the  movements  of  the  air.  Now,  when  rh}1:hmic 
changes  of  density  occur  in  a  middle  ear  whose  tympanum  is 
destroyed,  there  must  naturally  occur  a  back  and  forth  move- 
ment of  the  air  in  the  air  passage,  just  as  in  the  neck  of  a 
bottle.  These  back  and  forth  movements  of  the  air  may 
cause  by  friction  corresponding  movements  of  the  hammer 
and  anvil  and  thus  of  the  stirrup.  No  doubt,  stirrup  move- 
ments which  are  caused  in  this  way  must  be  of  small  magni- 
tude. But  no  one  who  knows  the  surprisingly  small  amount 
of  mechanical  energy  which  is  sufficient  to  call  forth  a  response 
of  the  auditory  organ  will  deny  that  they  might  result  in  an 
auditory  sensation. 


154  UNIVERSITY  OF  MISSOURI  STUDIES 

If  not  only  a  part  or  the  whole  of  the  tympanum  is  de- 
stroyed, but  the  chain  of  ossicles  is  also  lost,  the  mechanical 
processes  in  the  inner  ear  could  be  brought  about  by  pressure 
differences  on  the  two  windows.  An  air  wave,  coming  in 
through  the  external  passage  and  the  open  middle  ear,  would 
at  any  given  moment  affect  the  two  windows  with  a  slightly 
different  phase,  arriving  at  one  window  a  little  earlier  than 
at  the  other.  This  difference  of  phase  means,  of  course,  a 
difference  of  air  pressure  on  the  windows,  and  a  difference  of 
air  pressure  on  the  windows,  according  to  the  laws 
of  mechanics,  results  in  a  movement  of  the  internal  fluid 
from  the  point  of  higher  to  that  of  low;er  pressure.  It  is  plain, 
however,  that  this  difference  of  phase,  owing  to  the  small  distance 
between  the  two  windows,  must  be  very  slight ;  and  hearing  which 
results  in  this  way  must  be  rather  weak.  But  its  possibility  cannot 
be  doubted. 

Few  cases,  therefore,  will  be  found  where  a  sound  is 
heard  and  we  have  to  have  recourse  to  the  rather  improbable 
assumption  that  the  mere  passing  of  molecular  waves  of 
density  changes  through  the  head  and,  thus,  through  the  audi- 
tory nerve  ends  directly  results  in  some  weak  response  of  the 
nerves.  Nevertheless  at  least  we  may  admit  this  assumption 
as  possible.  To  admit  it  as  possible  would  not  cause  any  diffi- 
culty in  comprehending  the  ordinary  phenomena  of  audition, 
which  might  thus  seem  to  become  more  complicated  because 
such  density  waves  must,  of  course,  pass  through  the  head 
whenever  anybody  hears  anything.  But  such  effects  on  the 
nerve  ends,  granted  that  they  always  exist,  must  ordinarily 
be  overpowered  by  the  incomparably  stronger  stimulations 
simultaneously  received  by  the  nerve  ends  by  way  of  the 
stirrup  movement. 

Having  studied  the  function  of  the  human  ear,  it  is  in- 


MECHANICS  OF  THE  INNER  EAR 


135 


teresting  to  compare  this  with  the  organ  of  hearing  of  the 
lower  vertebrates.    Figure  42  indicates  the 
Comparative  manner  of  evolution  of  the     cochlea.     An 

anatomy  of  the  original  pit  (Fig.  43  a)  as  found  in  a  frog  is 
auditory  organ  gradually  lengthened  and  assumes  in  the  birds 
a  banana-like  shape  (Fig.  42  b),  showing  a 
distinct  tendency  to  coil.  In  mammals  the  process  of  lengthening 
and  coiling  has  proceeded  so  far  that  the  organ  (Fig.  42  c),  if  it 
were  transparent,  would  appear  as  a  spiral.  It  is  clear  that  the 
coiling  can  have  little  influence  on  the  mechanical  function 
of  the  organ.  The  lengthening  of  the  organ,  however,  is  of 
the  utmost  functional  importance.  The  original  pit  does  not 
differ  materially  from  the  other  cavities  which  we  find  within 


Fig.  42.     Evolution  of  the  auditory 
organ 

the  labyrinth,  communicating  with  the  semicircular  canals. 
In  this  pit  movements  of  the  fluid  caused  by  movements  of 
the  stirrup — or  rather  columella  plate,  since  the  lower  verte- 
brates have  a  much  simpler  connection  of  tympanum  and 
oval  window — produce,  probably  by  mere  friction,  stimulation 
of  the  endings  of  the  auditory  nerve.  The  organ  of  the  birds 
must  function  more  nearly  like  the  human  organ,  excepting 
the  difference  of  function  resulting  from  the  fact  that  the  endings 
of  the  auditory  nerve  are  spread  out  over  a  small  linear  extent, 
whereas  in  the  mammals  they  are  distributed  over  a  long  distance. 


136  UNIVERSITY  OF  MISSOURI  STUDIES 

In  birds  one  can  hardly  speak  of  some  nerve  ends  being  farther 
away  from  the  windows  than  others. 

It  is  of  some  interest,  in  this  connection,  to  note  that  ani- 
mals with  a  short  tube,  as  the  birds,  do  not  possess  in  the  par- 
tition of  the  tube  the  pillars  of  Corti.  They  can  get  along 
without  these  pillars.  And  naturally.  The  longer  the  tube, 
the  greater  is  the  maximum  pressure  which  may  act  upon  the 
partition  near  the  windows,  in  case  the  bulging  of  the  partition 
is  forced  to  proceed  far  towards  the  end  of  the  tube.  The 
greater  the  possible  pressure,  the  greater  is,  of  course,  the  need 
of  a  skeleton-like  support  in  order  to  protect  the  sensitive  cells 
from  collapsing.  Thus  the  mammals  need  the  pillars  because 
of  the  greater  length  of  the  tube. 

W'hat  must  be  the  difference  of  sound  perception  resulting 
from  these  anatomical  differences  in  various  species  of  ani- 
mals? We  saw  that  the  human  ear  can 
ComDarative  perceive  several  tones  at  the    same  time  be- 

psychology  of  cause  the  linear  extension  of  the  auditory 

the  sense  of  organ  permits  the    compound    mechanical 

hearing  processes,  transmitted    from    the    stirrup    to 

the  fluid  of  the  cochlea,  to  be  analyzed  into 
much  simpler  mechanical  processes  taking  place  in  successive 
sections  of  the  partition.  It  is  plain,  then,  that  in  the  auditory 
pit  of  a  frog  no  analysis  is  possible.  The  result  must  be  that 
the  frog's  ear  can  perceive  only  one  tone  at  any  moment;  and 
this  tone  is  most  probably,  as  a  rule,  the  highest  of  the  sev- 
eral tones  heard  simultaneously  under  the  same  circumstances 
by  the  human  ear. 

The  bird's  ear,  as  we  have  seen,  is  intermediate  between 
the  frog's  ear  and  the  human  ear.  But  it  does  not  seem  very 
probable  that  even  birds  can  perceive  very  many  tones  simulta- 
neously. The  fact  that  birds  sing  is  no  indication  to  the  con- 
trary, since  their  song  does  not  consist — like  orchestral  music 
— of  simultaneous,  but  only  of  successive  tones.  Of  more  sig- 
nificance, in  this  respect,  is  the   fact  that  some  birds,   for  ex- 


MECHANICS  OF  THE  INNER  EAR 


137 


ample,  parrots,  are  able  to  imitate  human  speech  sounds. 
Speech  sounds  are  characterized,  according  to  the  present  state 
of  phonetics,  by  particular  groupings  of  tones  in  both  simul- 
taneity and  succession.  It  is  not  certain  that  the  rough  imi- 
tation of  human  speech  sounds  by  parrots  is  more  than  an 
imitation  of  the  successive  groupings  of  tones.  Granted  even 
that  the  birds  possess  the  ability  to  perceive  more  than  one 
tone  simultaneously,  the  anatomical  facts  would  make  it  prob- 
able that  this  ability  is  very  limited  in  comparison  with  the 
human  ear  which  perceives  the  most  varied  combinations  of 
tones  in  speech  sounds  and  in  harmonic  music. 

Let  us  now  briefly  look  back  upon  what  we  have  done. 
We  have  regarded  the  organ  of  hearing  as  a  long  and  narrow 

tube,  filled  with  a  practically  incompress- 
The  need  of  ^^^^  ^^^"^  ^"^  divided  lengthwise  by  an  im- 

experimental  perfectly  elastic  partition  which  is  the  seat 

data  of  the  auditory  nerve  ends.    We  have  found 

that  the  problem  of  determining  exactly, 
for  each  given  form  of  stirrup  movement,  the  mechanical  pro- 
cesses taking  place  in  the  tube  is  from  the  mathematical  side 
an  almost  hopelessly  complex  one,  made  still  more  difficult 
by  the  lack  of  data  concerning  the  mere  facts  of  hearing  as 
well  as  the  elastic  and  other  physical  properties  of  the  parti- 
tion. In  order  to  overcome  the  intrinsic  and  accidental  diffi- 
culties standing  in  our  way,  we  have  introduced  six  simpli- 
fying provisional  assumptions ;  not  using  all  six  in  every  case, 
but  now  some  of  them,  now  others,  according  as  the  purpose 
of  the  moment  seems  to  warrant.  We  have  thus  obtained  a 
superficial,  but  for  a  beginning  satisfactory,  insight  into  the 
wonderful  machinery  by  which  we  analyze  the  complicated 
sound  waves  with  a  result  which — for  example,  with  respect 
to  the  hearing  of  difference  tones — is  most  surprizing  to  one 
who  knows  nothing  of  the  mechanics  of  the  inner  ear. 


138 


UNIVERSITY  OF  MISSOURI  STUDIES 


The  theory  thus  developed  does  not  pretend  to  be  the 
ultimate  solution  of  the  problems  attacked.    We  do  not  pos- 
sess the  data  upon  which  to  found  a  final 
theory.     But  we  shall  scarcely  obtain  these 
The  necessity         data     without    the     guidance     of    a    the- 
ory.       Experimental     research     must     be 
systematic,  must  start  frorh  a  theory,  how- 
ever imperfect  this  may  be,  in  order  to  lead  to 
scientific  advancement.     If  the  theory  here  offered  succeeds 
in  stimulating  experimental  research  in  a  field  somewhat  neg- 
lected for  many  years,  the  author's  hope  will  be  realized. 


of  a 
theory 


APPENDIX 

A  list  of  former  publications  by  the  same  author  concerning  the  me- 
chanics of  the  inner  ear: 

Uber  Kombinationstone  und  einige  hierzu  in  Beziehung 
stehende  akustische  Brscheinungen.  Zeitschrift  fur  Psychol- 
ogic   und    Physiologic   der    Sinnesorgane    11,     l<77-22t9.     1896. 

Zur  Theorie  dcr  Differenztone  und  der  Gehorsempfin- 
dungen  iiberhaupt.     Ibid.     16,  1-34.     1898. 

Uber  die  Intensitat  der  Einzeltone  zusammengesetztcr 
Klangc.     Ibid.     17,  1-14.     1898. 

Uber  die  Funktion  des  Gehororgans.  Verhandlungen  der 
PhysikaHschen  Gcsellschaft  zu  Berlin  17  (5),  49-55.     18i98. 

Zur  Theorie  des  Horcns.  Archiv  fiir  die  Gesammte  Physi- 
ologic 78,  346-362.     1899. 

Karl  L.  Schafer's  "Neue  Erklarung  dcr  subjectiven  Com- 
binationstone."     Ibid.     81,  49-60.     1900'. 

E.  ter  Kuile's  Theorie  des  Horcns.     Ibid.     81,  61-75.     1900. 

Zur  Theorie  der  Gerauschempfindungen.  Zeitschrift  fiir 
Psychologic  und  Physiologic  der  Sinnesorgane  31,  233-247.    1903. 

Uber  Kombinations-und  Asymmetrietone.  Annalen  der 
Physik  (Vierte  Folge)  12,  8^9-892.    1903. 

The  significance  of  wave-form  for  our  comprehension  of 
audition.     American  journal  of  psychology  18^   170-176.     1907. 


139 


INDEX 


Analysis,   37. 

Anatomy  of   the   inner  ear,    14,   69, 

81,  87. 
Beats,   55,   58,  66,   107,    115,   118. 
Birds,  5,  136. 
Brain,  23. 

Clay  experiment,  8. 
Cochlea,    1. 

Comparative  anatomy,  135. 
Comparative  psychology,  136. 
Computation,  24,  28. 
Corti's  organ,   16,  22,  136. 
Difference  tones,  37,  59,  68,  84,  123. 
Disappearance  of  higher  tone,  84,  122. 
Duality  of  the  organ,   131. 
Ear,  1,  14. 

Elasticity,  12,  18,  20,  87. 
External  ear,  1. 
Fluid  displacement,  76,  90,  95. 
Graphic  methods,   28,   39. 
Inflection  points,  85,  89,  92. 
Inner  ear,  14. 
Intensity,  32,  34,  42,  68,  77,  79,  96, 

122,  129. 
Labyrinth,  3. 
Leather  seated  chair,   12. 
Leverage  of  the  ossicles,  5. 
Mean  tone,  56,  58. 
Middle  ear,  3. 
Ossicles,  5. 
Overtones,  31. 


Partition,   11. 

Pathology,  132ff. 

Phase,  35,  44,  47ff. 

Pillars  of  Corti,  16,  136. 

Pinna,  1. 

Pressure  on  partition,  88. 

Provisional  assumptions,  25,  33,  69, 

87,  95. 
Reissner's  membrane,  21. 
Resonators,  19. 

Psychological   observation,    82,    137. 
Safety  valve,  14,  22. 
Sensitiveness  of  the  ear,  83. 
Snail,   1. 

Sounding  bodies,  32,  130. 
Stirrup,  4. 

Subjective  tones,   37,   59. 
Tension,   19. 
Tone   combinations : 

2  and    3 ;  35,  44,  48,  49. 
24  and  25;  50,  58. 

4  and    9;  62,  77. 
1  and     2;  85,  93. 

5  and     8;  96,  103,  109,  113, 

118,  122. 

3  and     8;    124. 

Tone  intensity,   32,   34,   42,    68,   77, 

79,  96,   122,   129. 
Tympanum,  4. 
Weber's  law,  129. 
Windows,   3,  6. 


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UNIVERSITY  OF  MISSOURI  STUDIES 


YOLT  ME  I 


Contribntions  to  a  Psychological  Theory  of  Music,  by 

Max  Meyer,  Ph.  D.,  Professor  of  Experimental  Psychol- 
ogy,    pp.  vi,  So.     1901.     75  cents.      Out  of  print. 

Origin  of  the  Coyenaiit  Vivien,  by  Raymond  Weeks,  Ph. 
D,,  P.ofessor  o"^  Romance  Languages,  pp.viii,  64.  1902. 
75  c  ;nts.      Out  tf  print. 

Tlie  EYoltttio'i  of  the  Northern  Part  of  the  Lowlands 
of  Southeast  ]!iissouri,  by  C.  F.  Marbut,  A.  M.,  Professor 
of  Geology,     pp.  viii,  63.     1902.     $1.25. 

JDileithyia,  by  Paul  V.  C.  Baur,  Ph.  D.,  Acting  Professor 
or  Classical  Archaeology,     pp.  vi,  90.     1902.     $1.00. 

Tlie  Kiarlit  of  Sanctuary  iu  England,  by  Norman  Mac- 
U^REN  fRENHOLME,  Ph.  D.,  Professor  of  History,  pp, 
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TOLl^lE  II 


1.  Ithaca  or  Leucas?  by  William  Gwathmky  Manly,  A. 
M.,  Professor  of  Greek  Language  and  Literature,  pp,  vi, 
153,     1903.     $1.00. 

2.  Public  Eelief   and   Private   Charity    in    England,    by 

Charles  A.  Ellwood,   Ph.   D.,  Professor  of  Sociology, 
pp.  viii,  96.     1903.     75  cents.      Out  of  print. 

3.  The  Process  of  Ir.dnctive  Inference,  by  Frank  Thill y, 
Pii.  D.,  Profe-^-or  oi  Philosophy,  pp.  v,  40.  1904.  .,5 
cent":. 

4.  RegfMieration   of  Crayfish    Appfjidages,     by    Mary    I. 

Steele,  M.  A.     pp.  viii,  47.     1904.     75  cents. 

5.  The  Spermatogenesis  of  Anax  Junius,  by  Caroline  Mc- 
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